I think your problem with UD (argument 1, in your second link) arises entirely from the way you choose to think about possible worlds. You built on a bad foundation, discovered the foundation was shaky, and so abandoned the original plan. But the problem was just the foundation, not the plan.
Both common sense and physics talk about the world as consisting of things-with-states. This remains true for possible worlds. Possible worlds defined using everyday concepts (e.g. worlds where "McCain defeated Obama in 2008") or using some exact physical theory (e.g. a billiard-ball world) still have this attribute. If you were to talk about all the possible billiard-ball worlds, there's no problem telling them apart, and it's easy to ask whether there's a natural measure on the set of such worlds.
But at your second link you write
There is an infinite number of universal Turing machines, so there is an infinite number of UD. If we want to use one UD as an objective measure, there has to be a universal Turing machine that is somehow uniquely suitable for this purpose. Why that UTM and not some other? We don't even know what that justification might look like.
So you've adopted a concept of possible world which is something like "possible program for a universal Turing machine". But the problem here is arising entirely from your idiosyncratic concept of possible world.
What does a universal Turing machine look like, from the things-with-states perspective? Consider the primordial example of a UTM, Turing's example of a tape moving back and forth through a read-write head. There are two things with states: the head and the tape. They undergo causal interaction and change states as a result.
Originally Turing was thinking of physical machines like the ones around him, made of metal and electronics and so forth. But suppose we try to take the UTM he described to be a universe in itself. How far can we go in that direction? Again, we can do it, thinking in terms of things-with-states and their interactions. We can think in terms of fundamental entities which have states and which can also be joined to each other in some sense. The tape is a one-dimensional string of entities joined side by side. The head is another entity which interacts with the entities making up the "tape", and whose join relations are also dynamical - it moves up and down the tape.
This all describes a type of possible world, just as the "billard-ball world" of elastically colliding impenetrable spheres in n-dimensional space is also a meaningful type of possible world. The dynamical rules for the Turing tape are the "laws of physics" for this world, each set of initial conditions gives rise to a possible history, and so on.
Now suppose you consider a different set of laws for the Turing-tape world. It still has the same structure, but the states and how they change are different. Is this mysterious? No, you've just defined a different class or subclass of possible worlds. Both classes of world are "computationally universal", but that doesn't mean that the world from one class which performs a particular computation is the same world as the world from the other class which performs that computation.
Yet this is what you're assuming, more or less, when you talk about having to pick a UTM as the UTM, in terms of which possible worlds will be defined. You're treating a possible world as a second-order abstraction (equivalence class of computations) and trying to do without a thing-with-states foundation. If you insist on having such a foundation, this problem goes away. You still have the very formidable problem of trying to enumerate all possible forms of interactions among things-with-states. There is still the even larger problem of identifying and justifying the broadest notion of possible world you are willing to consider. What about worlds where there's no time? What about worlds where there's no "physical law" - changes happen, but for no reason? But your particular problem is an artefact of computational idealism, where reality is supposed to consist of computational or mathematical "entities" which exist independently of anything like "things" or "substances".
I've collected some tips and tricks for answering hard questions, some of which may be original, and others I may have read somewhere and forgotten the source of. Please feel free to contribute more tips and tricks, or additional links to the sources or fuller explanations.
Don't stop at the first good answer. We know that human curiosity can be prematurely satiated. Sometimes we can quickly recognize a flaw in an answer that initially seemed good, but sometimes we can't, so we should keep looking for flaws and/or better answers.
Explore multiple approaches simultaneously. A hard question probably has multiple approaches that are roughly equally promising, otherwise it wouldn't be a hard question (well, unless it has no promising approaches). If there are several people attempting to answer it, they should explore different approaches. If you're trying to answer it alone, it makes sense to switch approaches (and look for new approaches) once a while.
Trust your intuitions, but don't waste too much time arguing for them. If several people are attempting to answer the same question and they have different intuitions about how best to approach it, it seems efficient for each to rely on his or her intuition to choose the approach to explore. It only makes sense to spend a lot of time arguing for your own intuition if you have some reason to believe that other people's intuitions are much worse than yours.
Go meta. Instead of attacking the question directly, ask "How should I answer a question like this?" It seems that when people are faced with a question, even one that has stumped great minds for ages, many just jump in and try to attack it with whatever intellectual tools they have at hand. For really hard questions, we may need to look for, or build, new tools.
Dissolve the question. Sometimes, the question is meaningless and asking it is just a cognitive error. If you can detect and correct the error then the question may just go away.
Sleep on it. I find that I tend to have a greater than average number of insights in the period of time just after I wake up and before I get out of bed. Our brains seem to continue to work while we're asleep, and it may help to prime it by reviewing the problem before going to sleep. (I think Eliezer wrote a post or comment to this effect, but I can't find it now.)
Be ready to recognize a good answer when you see it. The history of science shows that human knowledge does make progress, but sometimes only by an older generation dying off or retiring. It seems that we often can't recognize a good answer even when it's staring us in the face. I wish I knew more about what factors affect this ability, but one thing that might help is to avoid acquiring a high social status, or the mental state of having high social status. (See also, How To Actually Change Your Mind.)