Can I use a deterministic NONCE for AES-GCM file encryption if I generate "fresh" keys for each encrypted file
Yes, you can set the IV to zero if you use a different key each time. You speak about a random key, but is the key truly random, even in time? What happens if a file is changed, what attack vectors would surface in such a case? 

Using a 32 bit random "fixed" integer is not a good idea. You'd quickly produce collisions even for a relatively small amount of files stored with the same key. So whatever you try to fix with that, it won't be fixed enough to be called secure in a cryptographic sense.

If you worry about the keys not being random, I'd add some startup tests for the random number generator instead. Because without a trusted random number generator a lot of operations are not going to be secure.