For what I understand, Bitcore uses bitcoind through RPC. In particular it uses the module bitcoind-rpc. You can find it here: https://github.com/bitpay/bitcoind-rpc
As such, I think you could replace it with a different client which provide a similar RPC server.
However, I'm a little bit doubtful about how feasible it would be to replace bitcoind.
My idea is that Bitcore sources make use of many bitcoind data.
You can verify it by searching "bitcoind" in the bitcore-node repository.
In conclusion, I believe that it would be possible to do what you want but it would require a notable effort;
hence it may be not be worth doing it.
However, if the client is a fork a Bitcoin Core maybe you have a chance!
First you should check if all the data used from the bitcoind folder in the Bitcore sources are still present in the fork client. If so, I would make a try