No Mini can run OS 9 natively.
Your current TiBook might not be dead-at least on older(VGA) models, in my experience, a dead but connected PRAM battery can keep them from booting. Follow the iFixit guide for takedown, but the PRAM battery is under the optical drive. You don't even have to remove it-just try disconnecting it from the logic board and see if the computer will start.
Failing that, within your criteria your best bet is going to be another TiBook. I'd suggest looking for a 1ghz model, as these are the best and last laptops that can boot OS 9 natively. These can also officially run Leopard(although any TiBook can run it without a lot of extra trouble) and actually run it fairly well(max the RAM to 1gb).
A white iBook G3 would also work for you(I'd personally get a "snow" over a clamshell for what you're doing). I'm not familiar with the program you're using, but if it's not Altivec-enabled a G3 and G4 will effectively-clock for clock-be the same. The iBook G3s max out at 900mhz. The downside is that they can't hold as much RAM as a TiBook. They have some amount of memory soldered to the logic board(the amount varies depending on the model-I think it maxes out at about 128mb) and only one expansion slot. This means that the max they will hold is 512mb+whatever is on the logic board. The "snow" G3s and the last model(Firewire) Clamshells will run up to 10.4(Tiger).
Avoid iBook G4s and aluminum Powerbooks as none can run OS 9 natively.
If you can make room for a tower, I'd suggest either a Quicksilver or an MDD(FW400 or MDD2003). Many folks consider the single 1.25ghz MDD to be the best OS 9 machine ever made-and rightfully so as they are amazingly fast under OS 9. If you're running OS X also, though, I'd look for either a dual 1.0 or dual 1.25, as the second processor won't cause a performance hit under OS 9(some specific programs can even use it) and it does make a big difference in OS X.
iMac G4s don't have a huge footprint either, and some can run OS 9 natively(do your research on this!). One of these might be a good choice if you have the room for it. Plus, they are an iconic design and, IMO, one of the most ergonomic computers Apple has ever made.