Fein and things like Fein multicutters (Bosch do them in Blue and Green and corded and cordless varieties) would do the job, especially if its a reasonably short length of cut. The plunge blade is marked with increments rather having a stop though, so its a steady hand / eye thing
The blade vibrates almost imperceptibly to cut and is and is very safe to use - it'll cut wood and steel but not meat and fingers - its the same kind of saw the gets used for removing plaster casts. Different attachments can be used for sanding and grout removal too so a good general fixing/renovating/bodging diy buy.
Otherwise an angle grinder and a thin blade (such as the ones for stainless steel) and some care would work too.
If you surf the shopping channels theres are dinky-bladed plunge cutting saw that pops up there, but not sure how close to an edge they cut.
If its a longer cut then the festool and maffel type saws can cut very close to an edge (but not flush) and can plunge and have a depth stop, and will cut a very straight, clean line (unlike the multi cutter which can hack things up a bit) but thats in the hire rather than buy price range