I believe the H-S Precision stock has an aluminum bedding block.That will be CNC machined and should be a drop in fit for a factory spec Rem 700.
Some interference points have been mentioned,such as a safety.Good advice.I would think of you have the stock Rem trigger,the stock should have clearance.
I suppose there may be a variant of the H-S stock that is not aluminum bedded,but if someone has decided to do a glass bed over the aluminum(I have seen that mentioned in a post on tfl) they may have botched it.
I'd be leary if it looks glassed over.
Look to see you do not have a trigger pin or something walked out ,protruding.
There is a product,Prussian Blue,maybe your NAPA Auto parts will have it.Its about the consistancy of cold cream.and it is an indigo blue.Its purpose in life is to show when two parts touch.If one part has blue on it,it will transfer to the other part.
If you use an acid brush or pipe cleaner to apply a thin coat of prussian blue to your bbl/reciever,assemble it to the stock,and remove it,you will be able to read the spots and see what touches.