Will prepositional verbs always prevent us from moving the preposition to the middle of a sentence?

It actually depends on the context of the sentence. In everyday spoken English example 1 is fine.

In professional/academic written English example 3 is, in my opinion, the best choice.

I agree that example 2 is incomplete.

You can always arrange the phrase so that the preposition is not at the end of the sentence, although you have to be the judge of whether the result sounds fine or overly complicated.

In fact, many would consider sentence 3 the only "correct" option, because it does not end with the preposition. In reality, the usage in #1 is completely fine, just not very formal.

Sentence 2, is ungrammatical, though. You can't just get rid of the preposition, because it then becomes the simple past tense of to engage. It would have to be something like "...the project you are currently engaging."

Number three is the only one that is acceptable when putting something in writing. Number one is not only okay when speaking, but is probably better, as it sounds more casual and less stuffy.