Sorry about taking so long to reply. I have seen planes both re-mouthed and re-soled and have done both depending on the plane. The challenge with re-soling is in getting the new section of the bed to match up with the existing bed. I used hard maple and it worked out well. Re-mouthing involves letting in a wooden patch to close up the mouth and is a more traditional fix.
I wouldn’t try to apply a laminated piece to the side. I would just re-glue what is there, carve in and apply an oversized patch and cut it back to match the existing contours of the body. This is probably a project you would want to wait on until your knowledge and skill have advanced some, however it would be a project you would gain a lot of experience from.