That's funny, a couple of years ago I thought about modding my 424mkII, I bought parts and took it apart then left it sitting in a box.
I don't know what is different about mkIII (I thought that had phantom power?) Ask on the Analog Only or Tascam boards (which are pretty much the same board

) I don't think you'll find a shop to do affordable mods; if you're not up to it, maybe somebody here or elsewhere in DIY-land can help (I can't).
What I remember from mkII: It used 8 pin SIP chips, which are pretty much obsolete. I scored the some of the last stock of 5534 (or 32, can't remember) Mouser had before they ran out. I remember thinking that chip was an upgrade from whatever was stock. So your first task will be locating a supply of the appropriate chip, or using an adaptor board to get a more modern package to fit.
The mkII had no phantom and thus no high-voltage rail. I was gonna tap its positive rail (+16V I think) and use that. No, it's not +48V, if you want that you'll have to go back to the transformer and build a voltage multiplier. But +16V will power a lot of electret condensers, like the Shure SM81, many AT mics, and so forth. So it's worthwhile, I think, and requires a minimum of circuit bending--I needed to cut a couple of traces around the XLR input board, and run a wire and some resistors there. Mind you, I never did that, just planned it.
I was gonna do channels 3-4 as phantom and leave 1-2 unpowered. No particular reason, you could power all of them if you like.
More clean gain? Well, more gain is a function of the opamp feedback network. You can probably change a resistor for more gain, the trick is figuring out which resistor

I imagine somebody has a schemo somewhere. I don't remember this unit being particularly noisy, and a quieter opamp will help that.
An insert is simple, just get a switching jack like Neutrik NYS212, and tap the channel right after the preamp. Finding a place to mount the jack on the case will be the trickiest part; it's tightly engineered inside.
Overall, the build quality of the internals I thought was very good.