Yes, as long as nothing on the trailer requires something that the 13 pin would give then you’re absolutely fine. For the trailer, I suspect all you’ve got is the standard lights which will work perfectly well on 7pin.
All you need to do is just connect up ground and each of the individual lights that you need and you should be good to go. The only difference between the 7 and the 13 pin is to provide power for things that probably only caravans and more advanced trailers have
I have converted a relatively simple trailer and bike carrier from 7-pin to 13 pin as that’s what my car has and it was no hassle and no issues at all
You will find pinouts online as well as with more expensive adaptor plugs. They will come with a pinout diagram and all you need to do is just match what you need with what you’ve got.