Your team has never owned a lathe before. I’m gonna assume you also do not have an experienced machinist either or you likely wouldn’t be asking your questions. Forgive me if I’ve assumed too much. So with that in mind…
I’d suggest not buying an entry level machine. You’ll outgrow it pretty fast.
I’d suggest not buying a Rolls either. You can better spend the money on tooling and accessories.
So, Mama bear is best for FRC, IMHO. Without going into specific brands or models, I’d look for one that that can accommodate a collet closer (“Royal” collet closer style). This allows for a stop to be used in a collet and repetition of parts is much easier. …You’re always gonna be making multiple widgets. A collet (and collet closer) will greatly increase your efficiency.
A collet set and closer infers a thru hole in the head to be at least 1.125" diameter. That will determine the minimal lathe size (swing). Bed length will vary. A lot.
You can buy a lathe with a thru holes less than 1.125" but that won’t maximize the efficiency of the 5C collets.
You can look into a full collet set or just the major fractional sizes you work with. The larger the set, the more it cost.
Be sure to purchase collets that have a threaded internal hole. These accommodate threaded stops which work better than friction stops in the non-threaded hole version. …the friction stops can be used in a threaded hole, but I wouldn’t buy one in the first place. …remember, the outer diameter of the tail end of the collet is always threaded for whatever device closes that collet (Royal collet closer or other method).
The three jaw chuck should have an adjustable backing plate so you can dial in the concentricity semi-permanently. Yes, a four jaw can do this, but you’ll be doing it with every part. Inefficient. The three jaw should also have bolt on jaws so you can replace the factory steel jaws and also use machinable soft (Aluminum) jaws. …learn how to machine soft jaws on both your lathe and mill.
A steady rest is also useful for FRC parts. It’ll help stabilize long parts. A back burner project for me is to replace the standard bronze guides with bearing. I’d recommend that too.
Use a quick change tool post and holders (Aloris or clone). I prefer the wedge lock style over the cheaper piston lock style (just the way I was taught and used most of my career). A quick change tool post will allow you to, well, quickly change tools! Say: face the part, change tools to turn a diameter, change again to part off. You can easily keep repeating process cuz, remember, you’re gonna be making more than one widget.
DRO are cheap these days, but don’t skimp too much. You want one that will accept multiple tool offsets. …to take more advantage of your quick change tool post.
Shouldn’t be an issue, but ensure there is a diameter reading as well as radii reading. You’ll really want the X axis to read in diameter. Once you have the diameter turning tool set up, you can cut/turn directly to a specific diameter. Radii needs some arithmetic and is thus a place for silly mistakes.
If you mount the slides yourself, ENSURE they are parallel with the axis travel.
If a DRO is out of budget, 2" travel dial indicators with magnetic bases will work for most FRC parts. …I have a setup like this at home with my POS lathe. Trav-a-dial is another option. But by the time you buy one for each axis, you’re getting close to cheap DRO pricing.
Grinding HSS tools is a learned art. Not the hardest thing to do, but you’ll also need a good grinder and wheels with a decent wheel dresser. Good understanding of how the tool cuts and pressures the part is useful. Learning to grind in “chip breakers” is helpful. So, I’d recommend replaceable carbide inserts. Significantly higher up front costs, but faster and easier in a FRC environment.
If you’re buying inserts, then stay away from the brazed-on carbide “sets”. Those need a diamond wheel to sharpen. You can use a stone designed for carbide, but it’s a lot harder to actually get the edge you’ll want. Actually, you’d probably need the stone for roughing and the diamond for honing the edge.
Look for inserts that have a high rake angle, are “dead sharp”, and are polished for the free flow cutting of aluminum and plastics. Most inserts you’ll buy have a lower rake and larger edge radius for strength and durability on steel, stainless, etc. You’ll need some of those inserts too, but I’ll bet the majority of what your cutting is non-ferrous.
All kinds of new coatings are available for inserts. You can let the marketing folks sell you the proper one for your application. But for a non production environment (like FRC bots), I don’t see the need to pay for anything other than the high polish. Others will likely disagree.
PSA: Tie long hair back AND up. No long sleeves. No draw string (on hoodies, etc.). No dangling ear rings/piercings, necklaces. Learn to control a manual lathe (X, Z handles) when NOT standing in the path of thrown chips (stand off to the right side a bit).
DO NOT extend stock out the back of the head. Do I really need to mention safety glasses with side shield?
I know I didn’t answer your question about what lathe to buy, but I believe almost any lathe that meets your size requirements will work. I hoped to offer some of the after purchase questions.