I doubt MM and Tamiya thinner would be a mix made in heaven. Standard MM is a water based acrylic. When you thin a paint you're not just changing its viscosity - you're also changing the coherence of the paint itself. (That's why I think the chemists at Tamiya and Gunze have figured out how to allow their lacquers - whether they call them acrylics or not doesn't matter - to spray so well when heavily thinned with their lacquer thinner.) Tamiya A20 acrylic thinner is largely alcohol and many a good model has been made with it. I think good airbrushers can compensate for too much psi, thinner etc and thus they mistake their own skill for a good product. In fact Tamiya lacquer thinner works much better with Tamiya paints. (Tamiya or Gunze lacquer thinner might be poison with MM - it also might work like a charm. I'll try it out in the near future and report in.) So what you want is something that will thin the paint a bit and not break down its chemical composition leaving a weak or uneven coat. Frankly I've never used MM's own thinner but that's the obvious place to start. Vallejo is the archetype water based acrylic and I'd guess that their thinner (made for Model Air but works very well with Model Color) would very well with MM. I've had reasonably good luck with a homebrew of water and windex with Floquil acrylic (made by MM) - although that paint doesn't like alcohol. There's another option if you're willing to go off the beaten path. Artist acrylic fluids, made by many companies but the best by Golden, are thinned with something called "Airbrush Medium." This is liquid polymer and can be used to thin your coat considerably and, if anything, make it stronger. (For very high ratios like 5-1 thinner you'd also want another brew called Airbrush Extender. Works remarkably well.) The coat laid down initially is gloss and has a very smooth, almost waxy look to it. (Easy to get rid of the gloss at the end of a build numerous ways.) Personally I think that's what paint looks like when put on most metal surfaces - not the kind of look you get from a modern car where the finish looks like the metal has had its color changed. By the time you're done with weathering, the question is usually moot. If you go to the art store you might also see if they have a small bottle of GAC 200 which is a hardener. I've done little work with MM but heard it was vulnerable to peeling when masked. I painted the underside of a plane light blue reinforced with GAC 200 and it stood up to heavy masking very well. I think the artist stuff would work with any water based acrylic: no guarantees with the solvent based paints Tamiya and Gunze.
Whatever paint you use, keep a paintbrush and some water or thinner at the ready while you're spraying. Tip clogging can be a problem with acrylics and if you're quick it's most easily solved by simply wiping off the tip of the airbrush with a moistened brush. Do this every minute or so. It'll help a lot.