"detect opportunely" vs "opportunely detect"?

Neither works, actually. It would be better to rephrase the whole thing. Something like this might work:

Relatively few doctors work proactively to prevent [these] diseases by testing and monitoring their patients.

I would strongly recommend against putting "opportunely" or any other adverb inside that clause.

Only a few doctors assess their patients about the type and frequency of the tests they should be doing to detect and prevent such diseases, but we will help you solve this problem.

(As to the rule itself, writers disagree, but in general both orders are possible. See split infinitive.)

You didn't ask about the sentence in general, but it could definitely be shorter and clearer. Ralph M. has one good suggestion. Here's mine, a bit closer to the original:

Too few doctors assess the type and frequency of tests needed to detect and prevent such diseases, but we will help you solve this problem.