In my opinion it's just a different technique you have to learn.
I play a lot of fingerpicking on bass, like a classical guitarist would.
I often play classical guitar myself just for the sake of practicing finger strength, nuances, etc... all the wonderful things you can do on a classical guitar. (I'm not actually good at it, but I like classical guitar and performers who can play it well.)
So my fingernails I also keep in top shape, I suspect much like yours.
When playing on bass and I want regular plucking, the flesh of my fingertips are pulling the strings.
It's just a different technique you have to learn.
In your case: digging in. (You'll hear that phrase quite often if you're going to hang around here.

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That's something I assume you never had to do on a classical guitar. (At least, not as severe as on bass guitar.)
You have to keep in mind that a bass guitar is not as delicate as a classical guitar.
Compared to bass, all you ever did was gentle touch on strings.
Try to approach bass playing differently.
Your plucking fingers need to go between the string, and under it. As where on your classical guitar you approach your strings more from directly above.
Also, try to keep your hand more flat and not hovering above the strings. Even try with resting your palm on the body.
If you play standing try to hang your bass lower... so the instrument hangs lower than what you are used to with the classical.
You'll be more comfortable playing this new instrument, electric bass, if your elbow is in not such a sharp angle as it is on a classical guitar when sitting down.
(Although, keep in mind this is my opinion and advice for what it's worth. A lot of bassist prefer to hang their bass high... even up to their chest. I personally feel this is not ergonomic.
I play metal... huge disclaimer also.

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And: practice, practice, practice....
my 2 cents
Welcome to the world of bass.
I've always been an afficionado of classical guitar.
IMO (in my opinion) when it comes to playing technique it's easy for a bassist to pickup classical guitar or vise versa in your case.