"Safe" non-cumulative poison?

We have two cats, once of which is an "excellent" mouser. I write
"excellent" because, while he is great at catching them, he is terrible
at killing them. He brought in a pregnant female field mouse from
outside, and then lost her, and now we have a mouse problem.

The last time we had a mouse problem (this was before we had our
mouser), nothing worked for weeks until I turned to poison. I laid
poison everywhe The ceiling, inside the crawlspace walls, under the
sink cabinets, and it completely took care of the problem in a few
days. My fear is that, if I lay out poison again, the cat will catch
and chomp on a diseased mouse and may die.

I read somewhere that there are "safe" poisons that are
"non-cumulative" and won't hurt the cat even if he completely eats a
mouse killed via the poison. Is this true? If so, what is a "safe"
poison I can use?

How have others dealt with this problem? Trapping is *not* an option;
we are overrun (they are coming out during the day, which is a sign
that you have a full-blown infestation).

Any and all advice and comments are welcome!

Your safest bet might be to call in a professional, and if necessary
board the cats with friends or family for a time. BTW--that "if" is a
key qualifier, because a professional might be able to offer the safe
option you are looking for--in addition to a guarantee of service.
Doing it on your own comes with risks of incomplete efforts along with
those to your felines.

Best of luck!