Hi Noah,
It can be fixed with some clear two ton epoxy and a clamp. Don't use that 5 minute crap.
I saw in the second photo that you have the part that broke out. Good.
It broke along the grain line so should fit back together very well if you don't mess with it.
Because the tailpiece is a part of the bridge I'll refer to it as the bridge.
You will need the two ton clear epoxy, two small pieces of soft wood for clamping cauls to protect the instrument, some alcohol, a small rag that you don't need to use again, a pocket knife or other scraping tool that is small enough to get into the tight places where the strings will be later & wax paper.
.1. Dry fit the parts together and clamp them as a test run first to make sure they all go together well.
.2. Assuming they all fit and the clamps will hold them go to the next step.
.3. The epoxy comes in two parts. Mix small equal amounts together well with a small stick on a business card you don't need. You can use the same stick to spread the epoxy on the broken parts where they broke.
.4. Press them together tightly with your fingers to be sure they will not slide around.
Some of the epoxy will squeeze out and some will get on your fingers as expected.
.5. Use the alcohol with a rag to clean away any squeeze out and your fingers. If it's grain alcohol you can drink a little to calm your nerves too.
.6. Use the wax paper between the soft wood clamping cauls* and the bridge parts.
*Clamping cauls are used to prevent the steel clamps from marking and indenting the guitar or bridge.
.7. After about 10 minutes the epoxy will start to harden. That is a good time to quickly take the clamps off, look for more squeeze out and if any is there scrape it off with a pocket knife. After 30 minutes the epoxy will be bonded to well to break apart, but it is not yet hardened as much as it will later.
.8. Put the clamps back to hold it all together tightly and leave it for 24 hours.
After that time take off the clamps, cauls & wax paper. If you missed some squeeze out you may need to use that small knife to remove the excess and be very careful here with this, but you should not have any. Let it sit another 24 hours to be sure the glues chemical reaction is complete.
It should be able to hold those lower strings that generally have less tension than the high strings.
If for any reason it does not hold you should contact Mr. Putman (Now Retired) and see if he has an extra part he can send to you?