Should I use "in any case" or "in either case" in this example?

My research wasn't immoral. The only difference was that they were doing it for the sake of the animals, and I was doing it for my own. In any/either case who is to say which reason was 'better'?

What's the right alternative? Or, maybe another connector is better suited for this situation?

"Any", as you seem to recognize, does give an implication that there are multiple options. However, "in either case" is used less frequently and is perhaps less appropriate or recognizable. "Either way" may be a better choice if you want to highlight that there were specifically two options.

This may be a situation where there is no need for transition. Your use of a comparative ("better") makes your intent clear. Also, it could be better to just remove the rhetorical nature of the sentence and make your opinion clear. Alternatively, allow the question to be less leading.

Neither rationale is strictly better.
Is one reason better than the other?

If you wish the sentence to remain rhetorical, I prefer a single-word solution:

Regardless, who can say which reason was better?

Here you can take the case "they were doing it for the sake of the animals" as the first case and the "I was doing it for my own" as the second case. So using either case would be a better option.

EDIT: Having re-read your question, I realized that the "any case" doesn't seem refer to the two reasons, but rather to some other (yet undiscovered) set of instances or circumstances. So I'd go with "any" in this case since the number of such "cases" is undefined.

ORIGINAL: Both are fine, but if there were more choices offered, "any" would be the choice, since "either" indicates exactly two choices. In this context, "any" will also have the effect of preemptively including any other options offered by your interlocutor.

"In either case" doesn't seem right here. You are not really making a statement which applies to either case A or case B, but comparing the two cases with each other by asking "which is better".

If you were making a statement about both cases, the meanings of "In either case" or "In any case" or are different: "In either case the animals died" is a statement of the fact that "in both pieces of research the animals died". ""I any case the animals died" adds another layer or meaning: the reason for doing the research was irrelevant to the fact that the animals died. In this example, phrases for more than two cases meaning the same as "in either case ..." are "In each case", "In every case ..." or "In all cases, ...", not "In any case ...".

In your example, "Anyway, who is to say..." would be an alternative to "In any case, who is to say..." More formally, "Notwithstanding, who is to say...". If you don't mind starting a sentence with a preposition, "But who is to say...". In very informal speech, "Whatever, who is to say...".