I'm afraid that Tom and Bill sound as strange to me as Joan and the unknown interlocutor.
But following on from Komountain's previous comment about parallelism:
1. I prefer to | wait | rather than | go.
— Parallel: the emboldened verbs both relate to the same subject.
2. I prefer to | do it myself | rather than | ask Joan (to do it).
— Parallel: as #1.
3. I prefer to | do it myself | rather than | have Joan do it.
— Parallel: again, as #1.
4. I prefer to ask | him | rather than | her.
— Parallel: the emboldened (object) pronouns bear the same relation to the same verb.
5. I prefer | to do it myself | rather than | Joan do it. ("do" = bare infinitive)
— Not parallel: first "object" of "prefer" is a verb phrase, the second a noun phrase. The first "do" relates to a subject; the second to an object.
6. I prefer | to do it myself | rather than | that Joan (should) do it.
— Not parallel: the first "object" of "prefer" is an infinitive phrase; the second, a "that" clause.
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1-4 seem fine to me. 5 is distinctly odd. 6 is odd too, but less so than 5.
I'd be interested to see #5 translated into other languages. Could we retain the structure, I wonder?
MrP