"Please write with a pen."
Yes, you are correct. What I meant to question though is that the て form connects つかう and 書く rather than indicate that the pen is the instrument used for writing. wouldn't this sentence be asking someone to do two things - use a pen, and write - that aren't necessarily related? for example, use a pen to draw a picture and then write a letter using a pencil.
Oh sorry, I didn't realize that was your question.
Anyway, as far as I know, no, that's not what it's asking. When the て form of つかう connects directly to the て form of 書く, it is at least implied that you use the pen to write, as a single action. I think that if the speaker meant to request that you do two different things, then they would make it clear. I think your example would be translated into ペンで絵を描いて、そしてえんぴつで手紙を書いて下さい。Or something like that.
Your original example of ペンで書いて下さい is what I would use, though. But while that would be my choice, I think duolingo's example/translation is correct, just a different way of saying it.
Honestly, I'm not sure if there is a rule of some sort to make this so, and there might be exceptions as well, so you might get a more detailed answer from someone else. The only reason I know what it's asking is because I attended a Japanese school a few times, which forced me to learn how to be like them. I forgot most of it, but I still remember some.
I agree its pretty awkward thinking in English "please write with a pen" --> 「ペンで書いて下さい」but only having the option to make the valid but less equivalent sentence 「ぺンをつかって書いて下さい」--> "Please use a pen to write." One thing here is the emphasis landing else where (please use a pen to write vs. Please write a pen if that makes sense.) The lengthier sentence implies less routine instruction and just. Doesn't match the given English.