j'ai dû, je dois

This is similar to your previous question. 

https://progress.lawlessfrench.com/questions/view/ca-a-du-etre-vs-ca-doit-avoir-ete

As I said then, I think it should be acceptable, but perhaps Cécile could confirm this.

I think the two sentences speak of events in different times.

J'ai dû le lire trop vite. -- I must have read t too fast. (An event in the past)
Je dois l'avoir lu trop vite! -- I must have it read too fast. (A statement in the present.)

I think it is deceiving that the two English versions sound almost the same. Cécil's input would be appreciated.

Hi Chris,

If you look at the other question I mentioned, I linked to a discussion on wordreference which has some input from a native French speaker.  The example discussed in that thread is almost the same as the one in this question.

By the way, "I must have it read too fast" doesn't work in English.

Hi Alan,

yes, I realize the English is a bit warped. I intended the sentence to be parsed according to "I must have my Martinis shaken, not stirred". If you make the substitution of "shaken" --> "read to fast", you'll see what I mean.

OK, I understand what you mean, although it would be odd to insist on something being "too fast", since that's implicitly bad. But perhaps you think the French sentence is also odd.

Did you read the wordreference thread? It explains that both constructions can describe either an obligation (in the past or future), or a supposition about the past. In this context, it's the supposition about the past that's relevant.

Yes, I read the wordreference thread. Interesting.