My apologies if I'm repeating anything that's already been said; I've been isolating myself from the online discussion to attempt the Final Exam by myself. So here's what I've got:

I'm pleased with myself for coming up with the first two solution classes I've listed (and vague ideas about the third, which I later separated into its own class) within 60 seconds, but I didn’t come up with the full Obvious Solution in that time. More work to do...

Solution Class 1: Transfiguration

We know that Harry can Transfigure acids nasty enough to instakill when used to replace a small cross-section of a brain, because this is what he did to the troll (Chapter 89). His ability to perform partial Transfiguration in general is by now well-established. We also know from Chapter 104 that Harry can control how a Transfiguring object approaches the completed Transfigured form, as demonstrated by him succeeding at the shaping exercises he was doing during the Quidditch game. McGonagall also establishes in Chapter 15 that free Transfiguration is wordless and requires no wand movements (only contact between the wand and some part of whatever is to be Transfigured), so Harry can perform a Transfiguration without drawing fire from the Death Eaters by speaking incantations or making wand movements. This solution class involves Harry evading immediate death by combining partial Transfiguration with shaping control to kill the Death Eaters without drawing their fire or killing himself in the process.

In order to kill the Death Eaters, he can use partial Transfiguration to Transfigure a thin cross-section of their brains, much like he did with the troll. However, he will want to choose a cross-section that has a very small total volume for each brain so that he can affect all 36 Death Eaters simultaneously and still make the Transfiguration quick enough to be done before they can react. Some vital part of the brain stem (say, the medulla oblongata) will do nicely. If he wanted to be sure, he could also take a slightly larger cross-section and sever some of the major arteries nearby (supplying assorted parts of the brain) as well.

However, he must choose a volume to Transfigure such that some part of it touches his wand. It is unclear if the requirement is that the volume touches the tip of his wand or merely any part, but either case is solvable.

If he can feel whether a Transfiguration takes or fails, as he has been able to for other spells (such as when he tried to Finite and Alohomora the door in Chapter 23 after Draco locked him in), then he may safely test whether partial Transfiguration allows him to Transfigure air (he didn’t test this since discovering partial Transfiguration) by attempting to Transfigure a tiny volume of air near his wand into something safe (like steel) and feeling whether it takes. Because of the small volume, this test should take less than 5 seconds, not be visible to others, and he should be able to do other things while he’s at it. If he can’t feel whether a Transfiguration takes or fails, he shouldn’t attempt this test and should assume he can’t do so.

If he knows he can partly Transfigure air, he can choose a volume consisting of thin lines of air from the tip of his wand to each Death Eater, further thin lines through each Death Eater’s armor and clothing leading from the air to the chosen cross-section of brain, and the chosen cross-section itself in each Death Eater. He could also choose a volume consisting of a thin line of air from his wand to the ground, and then proceed from the ground using Transfiguration of solids (in case, say, gaseous Transfiguration is harder). (This also opens up the option to quickly Transfigure objects out of the air that will kill the Death Eaters, such as very thin blades, but I don’t know any materials offhand that would have the required compressive strength to make a very thin blade that can nevertheless behead a Death Eater fast enough. Carbon nanotubes might just break, although Transfiguring against tension may help.)

If he doesn’t know he can partly Transfigure air, he can choose a volume composed of solid objects connecting his wand to the desired cross-section. If the volume must touch any part of his wand but does not have to touch the tip, he can choose a volume consisting of a thin line through one of his fingers (preferably the pinky finger just in case something goes wrong) touching the wand up his arm, then down the outside of his torso and leg to the ground, further thin lines through the ground to each Death Eater, yet further thin lines from the ground underneath each Death Eater through their shoes, socks, and bodies to the chosen cross-section of brain, and the chosen cross-section itself in each Death Eater. If the volume must touch the tip of his wand, he can either grasp his wand by the tip without raising it or start the volume with a thin line along the outside of his wand from the tip to one of his fingers, then proceed as above with a volume composed of solid objects. (Recall from Chapter 91 that his wand is robust against small chemical burns.)

Finally, he can use his shaping abilities to Transfigure the desired cross-sections of Death Eater brains into acid before the Transfiguration takes effect on the remainder of the volume, then choose not to complete the Transfiguration of the entire volume (thereby avoiding damage himself). As a further backup, he can also choose a multi-part target form such that the remainder of the volume is Transfigured into something harmless (possibly even exactly what it was before) just in case.

If he can figure out which Death Eater is Lucius (through the code names, perhaps?) he might choose to spare Lucius and hope for his aid.

(Partial) Solution Class 2: Self-Sacrifice

Harry might reason that Voldemort gaining power and taking over the world is a disaster already set in motion by his hand, because he didn’t sufficiently question Quirrell / refrain from showing Quirrell the Hallows symbol / tell Dumbledore about the sense of doom / tell the Order what really happened in the Azkaban breakout, etc. and Hermione would consider Voldemort’s reign a catastrophe. Under the Vow, this would allow him to take lesser risks in order to stop Voldemort or even deal major setbacks to Voldemort’s plan.

So Harry might be willing to risk his own life in order to discarnate Voldemort (and maybe kill the Death Eaters). He could try casting on Voldemort, but that seems stupid in comparison to using the partial Transfiguration attack with the addition of Transfiguring the ground under Voldemort into antimatter (or possibly other suitable explosives if he knows them well enough to Transfigure, but antimatter has the advantages of needing very little Transfigured mass and not leaving any debris that will revert). In order to be sure of discarnating Voldemort, Harry will probably have to let himself be caught in the blast radius; the hope would be that newly enhanced Hermione will survive and maybe become a Light Lady. Alternatively, if Harry has some reason to know that casting on Voldemort would kill (rather than just discarnate) him, that might be enough reason to cast on him despite drawing fire from the Death Eaters.

This is not a complete solution class because it doesn’t let Harry evade immediate death, but it still seems better than death on Voldemort’s terms.

Harry might also notice that the Unbreakable Vow is mind-altering magic and consider it strong enough that he is no longer the same person from before the vow. This might reset his phoenix eligibility, so he might survive anyway on the tiny off-chance that Voldemort and the Death Eaters didn’t put anti-phoenix wards up.

(remainder in child comment due to length limits)