What printer should I consider for printing microfluidic channels?
"What Printer?" Let's look at your options:

FDM is the cheapest route into 3D printing, and may be cheap enough to buy just as a learning tool, as many of the concepts of 3D printing are fairly universal. It will not however likely have high enough dimensional accuracy for consistent 200 µm holes, and parts are not usually considered water tight (They may be most of the time, but pinholes are somewhat common).

BinderJet binder jet is the cheaper of the options for printing with powder, where basically glue is deposited onto layers of powder to solidify a part. I have personally used one to create a sand mould to cast metal into without having to create a positive with which to make the mould (ie lost wax casting). It can also be used to print ceramics or metals that are then fired in an oven into a more solid part. In general the dimensional accuracy is going to depend on not only the precision of the glue deposition, but the input powder as well. It may be possible, but I've never seen a water tight part come out of a binder jet.

SLM (SLS, DMLS, generally any metal printer with a laser (even sometimes an electron beam)) Can print quite high quality parts, but a 200 µm hole is pushing into the realm of difficult to achieve. Metal printers produce parts with a fairly rough surface (compared to a machined part), which is dependent on a lot of things, but primarily the input powder size distribution. Typical SLM printers use powder anywhere from 20 - 60 µm in diameter. With very small holes, you'll have to begin wondering about the likelihood that a hole will be sealed at some point by the random surface roughness exceeding the diameter of the hole. This also translates into a path with a lot of resistance, so pushing a fluid through it will require a lot of pressure. Again, I'm not in any sort of biological field, but I'm aware Titanium is pretty bio-compatible, and that is a *fairly commonly printed alloy (the current list of alloys that are pretty easy / standard to print isn't terribly long). Finally price: you can't afford it. I work with a relatively small (though somewhat high end) SLM tool at work, and I believe the final cost went up somewhere in the neighborhood of 500k.

SLA will likely be your best option. I don't have any experience with bio-compatibility (I'm a metallurgist), but there are many uv curing resin systems out there that are made for SLA printing. This is also likely the only type of system that can meet your dimensional requirements (water tight, small features, clear material, etc.) as well as your price target. Nothing can replace doing the research on your own, but this would be my choice. As for which SLA... (Note I'm fairly biased in this opinion) I'd look into Formlabs and possibly in particular their clear dental resin.

*PolyJet I don't know much about this one. It seems like a cross between SLA and FDM, but it seems likely to be out due to cost.