I'd like to put a filled backpack in my suitcase (for check-in luggage) along some other items -- this is to protect the items inside the backpack from damage.
Is this allowed/any other potential issues I should know about?
Yes. A bag is not a restricted item and so you can back as many bags in your suitcase or another bag.
This something I often do to carry specialty bags. I have used this to pack camera bags that I will use at my destination, backpacks to use for hiking or as day pack, beach bags, etc. My wife often packs some purses that match different outfits in too.
There two things to be mindful about with regards to security, since they can open bags for inspection:
As others have said, yes, it is perfectly fine to do so. Last time I did this was only a couple of months ago, coming from Tbilisi to London. I left London with two bags, as I was taking quite a bit of stuff there and had to pay for the second bag. On the way back I only had a few personal items, so put them in the smaller suitcase and put the smaller one into the larger one. It was still within allowed weight, so nobody asked anything.