I'm running a private blockchain and I've been using Metamask connected to my custom RPC URL with no issues.. I'm curious if there's any other Ethereum wallet that let's you add and connect to a custom RPC URL.
Currently there are not that many wallets that allow to connect to random networks. MetaMask tends to be the only one.
Your best bet is to try here https://walletconnect.org/wallets - also some of the wallets are open source, so you can add this custom functionality to these wallets yourself.