These are some fast guesses-- my impression is that it can take years [1] to track down this sort of thing. Also, I don't know how much of this has already been done.

Start with five minutes thought. What does Eliezer know about his symptoms? Can anything be deduced by mulling over them?

I'd start with poking around to find out whether other people have the same pattern of symptoms. Does it have a medical name? What does medical research say about what works? What do people say about what works? Do the symptoms ever become better or worse? Does this correlate with something that could be experimented with?

Hire MetaMed, but also look for anecdotal information.

Exercise might be bad for some people.

I'm going to recommend some caution about experiments-- so far as I know, Eliezer has fairly good health. He's got some energy problems, an inability to lose weight, reacts very badly to missing a meal, and doesn't get any good from exercise. There's a lot of room for making things worse.

I'm in substantial agreement with this, but I do think the bad reaction to missing a meal is enough to be of at least a little concern. On the other hand, the cultural issues around fat are weird and extreme enough that it could explain the lack of thought that's gone into Eliezer's efforts to lose weight.

[1] Something in the neighborhood of 2 years or more for people who report success. Original research takes time.