What Canaboo suggested is a good idea if there is room for a big enough slot while still leaving enough metal behind it.

Replacing the dropouts is not really doable without proper framebuilding skills and equipment which means torch brazing or TIG.

As a hack you could try extending the existing dropout backwards and then you can cut a bigger slot. The actual dropout is about 5mm thick so could be welded with MIG, flux-core or even stick. Grind down the back of the existing dropout, cut a curved shape out of a bit of 5mm plate (or whatever the thickness you have is), bevel both parts, and get a good weld in there. Grind it all flat again (so never mind a bit of spatter and boogers) then extend the dropout backwards with a grinder, dremel and/or file depending on your patience level. Finally trim off any of the new bit of metal you just welded on that you don't need and tidy it up. You want to leave a bigger piece on there while actually welding it to act as a bit of a heat sink. ​​​There's no reason why this couldn't work perfectly well if you make a nice job of it. But practice on a scrap Walmart frame before mutilating the nice cromoly retro MTB frame you just spent ages hunting down.