Only eaten about 3 times in 5 months

[Malum Argenteum]

What's the best method for, say, a small rat, which I think could be the right size for my BP?

Best, in my experience, is pressurized CO2 with a flowmeter at a rate of 30-70% of the rat's enclosure volume per minute. A 5lb CO2 tank and flowmeter can be had for about $125; a full tank at a local welding supply or beer CO2 shop, and a flowmeter off Amazon, is often the least expensive way to go. Contrary to the belief of many people, CO2 asphyxiation using dry ice is not humane, and it is not approved for euthanasia in the US.

Cervical dislocation (basically, breaking the neck) is also approved, though can be tough to master (I could not get the hang of it, anyway, and I'm fairly proficient at harvesting a range of animals for human consumption).

There are a couple other non chemical methods described in the AVMA guidelines, but probably not ideal for the situation. Best to confirm any non-professional advice (such as mine) against the AVMA guidelines before proceeding.

AVMA euthanasia guidelines.

Instructions for cervical dislocation.