Tile trim sounds like a good solution. Yup a palm router will make a neat cut so long as you guide it with a straight edge stuck to the surface[1]
Assuming the laminate is something absorbent, I would treat the cut edge to stop any chance of water getting in - say by applying couple of coats of varnish or painting with SBR and letting it sink in. (mask the visible face so you don't get any on that). I would probably add some clear silicone when bedding in the tile trim (to both fix it in place, and also exclude water ingress near the cut edge in future).
[1] To mount a batten for the router using masking tape on the face of the laminate, and on the back of the batten, then use a few drops of CA glue to glue the surfaces of the masking tape to each other. That will hold firm enough to route against, but you can swill prise it off easily when done, and not risk marking the surfaces.
A router novice should certainly go down the tile trim route. At least if you mess up on the workbench you only lose the workpiece, mess up in situ and you have to redo the whole job.
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