Do I have to repeat the subject of this sentence?

The sentence that's troubling me is:

Is he signed to any label or is he self-employed?

I don't know if it is necessary to put "is he" a second time, or if I can safely omit it. My tacit instinct tells me that it has to be there but I can't find any reason why it needs to be repeated.

I've further contrasted it with this sentence,

The treasure was hidden in the cave or in the underground lagoon.

which contains the same coordinating conjunction and clearly no repetition is necessary. So at first I thought perhaps it was that conjunction which made it necessary, but that seems to be wrong. Then I thought that maybe because it's a question it must be repeated. But that's also easily disproved by

Is he happy or sad?

Moreover, when I deconstruct the sentence it seems like (signed to any label) and (self-employed) both act adjectivally just as in Is he happy or sad? So, is it actually okay to say "Is he signed to any label or self-employed?" or am I missing something obvious?

'Is he signed to any label or self-employed?' is every bit as grammatical as 'Is he signed to any label or is he self-employed?' With sentences coordinated using 'or' (and other coordinators), the subject may (almost?!) always be omitted if ambiguity doesn't thereby arise, though it may be better to repeat it for ease of interpretation with long clauses etc:

'He went to see the old lady who lives in the small house near the spring under the lower wood, the one with all the larches, or he went to see the charming little old chapel.'

However, even where obvious ambiguity doesn't arise (here's an example where it does:)

It was very gloomy; what was it that he saw? He saw a fox with a wolverine or he saw a wolf.

It was very gloomy; what was it that he saw? He saw a fox with a wolverine or a wolf.

the default readings may not be the same.

"Is he Lithuanian or Latvian?" need not entail an either-or situation (... "His accent sounds like he might come from that general area."

whereas

"Is he Lithuanian or is he Latvian?"

defaults more clearly to an either-or foreknowledge / assumption.

..............

I'd say that the three examples OP gives are clearly either-or situations, so the repeat of subject is not needed to signal this, and the 'avoid unnecessary weightiness' rule assumes priority unless emphasis is an overriding consideration:

Is he signed to any label ... or is he self-employed?(!) (dramatic drum-roll)