I have a book for you to read. It's called Garden Bulbs for the South and is authored by Scott Ogden. Mr. Ogden lives in south Texas and though this book applies to all the southeast, it is particularly useful to Gulf Coast residents.

In short, I have never had rodents bother daffodils (Not all will do well in the south. See the above book.), crinums, or zephyranthes (rain lilies). All of these will do well for you there in New Orleans and none need to be lifted from season to season. I suspect you are thinking of tulips. These must be refrigerated for 8 to 10 weeks in the fall before planting in late December or January each year. Even if you lift them, they will only rarely ever bloom again even one more year.

In fact, Mammals will not feed on bulbs from the AMARYLLIDS.
Since the environmental conditions are as adverse as you indicate, I would consider to build a large, unheated greenhouse. One that is safe to keep out, even mice and such, that might invade via the soil around (!!!).
With such a facility, you can grow even the most precious Japanese Lilium hybrids (of L. auratum and L. speciosum) that those would fail in soggy substrate. And just, simple, Gladiolas and Hippeastrums will reveal their FULL beauty under STRICT protection from rain.
In Germany, rain for day after day is a CURSE, too, that outrages me, and I grow exclusively in containers (I am a specialist in this fashion of cultivating bulbs)

Might not work in your area - however, the jonquil daffodils naturalise readily, don't mind persistent rain over both winter and summer, tolerate some surface flooding and dig themselves down deeper than is usual for Narcissus, so they're likely to evade hungry creatures.

Paperwhites and Erlicheer would both be worth a try. And - don't dig them up. The clumps can persist for many years without suffering from overcrowding. They do appreciate a topping of good compost in autumn, however.