I think there was a mistake in choosing the segment to boldface.
"Your fate to be required to bear" is not itself a constituent. It's just a part of a constituent.
(Parenthetically, the answer to the presenting question is No; only some verbs -- transitive and intransitive, because infinitives can be subjects, too -- can take infinitive complements. Of course there are other kinds of infinitives, too, but this answer is already too long.)
The smallest constituent that contains this string -- and it still needs quite a lot of untangling, because it's been done plenty of things to -- is
- what it is your fate to be required to bear
the direct object of the higher bear.
This construction is an embedded question complement clause. The introductory what gets "moved" by Question Formation from (i.e, it appears somewhere other than) its normal position in the clause, which would be after the lower bear as its direct object.
(in the dissection that follows, I mark items that are moved or deleted as code)
- it is your fate to be required to bear
what
That, in turn, has been Extraposed from
for you to be required to bear what is your fate
That is, the subject of is your fate is an infinitive clause:
for you to be required to bear what
You is the subject of the infinitive verb be required.
Subjects are marked with for in an infinitive, and both for and you get deleted here, because infinitive subjects normally are deleted, either because they're indefinite and apply to everybody, or -- as in this case -- because they're predictable from elsewhere in the context, and you gets mentioned in your duty.
This clause, too, has been done things to. It's a Passive clause, so you has been "moved" by Passive from its position as the object of require. (What was the subject? Who knows? That's what indef means.)
for indef to require you to bear what
with another you as the object (or requiree) of the requirement.
And we're not done yet. There's still another infinitive, the lower bear, which is intended to be in parallel with the upper bear, and to reinforce the message.
It has a subject you, in fact the same you that got "moved" by B-Raising to be the object you in the require clause above.
This whole process is what I meant in this handout by "unwinding" syntactic rules, constructions, or alternations.
Now I have to go wash my hands.
Best Answer
Traditionally, a clause is indeed a finite verb and all its dependencies. The subject of the sentence is he, the (direct) object his daughter. The verb let is special in that it often has an object and an infinitive as a tertiary complement (third thingy that strongly depends on it, besides subject and object). You could analyse the infinitive after let as an object complement, because it is very much related to the object, his daughter.
An infinitive is externally much like a noun (it can be governed by a verb); internally it is a verb (it can have arguments that verbs can normally have). It has an argument that depends on it: to the music. One might call the latter an adverbial constituent.
Other linguists use a different definition of clause: they define it as any verb and its dependencies. In that case, the infinitive listen and its argument to the music form a subordinate clause together. It doesn't matter which definition you choose, as long as you are consistent.
But, even according to that definition, I wouldn't call his daughter a subject, because subjects are normally marked as such:
It does fulfil the semantic role that the subject normally has with the verb listen ("experiencer"), whenever there is such a subject; but subject is a syntactic category, not a semantic role, so that is irrelevant. If we called her the subject of listen, then we would have to do the same in this sentence:
Her expresses the same semantic role as she ("agent"), as is normal in passive sentences with by; but we never call her the subject of the verb in such cases, because semantic role is not what the term subject is all about.