I tried a bitcoin wallet…
There are many different Bitcoin wallets and the answer may depend on exactly which one you are using. For example, there is a “Bitcoin Wallet” for Android phones which is sometimes known as a Schildbach wallet. There is also a Bitcoin Core for computers and many other devices.
…but it only accesses active Bitcoin, not the funds in my wallet.
A typical Bitcoin wallet like Bitcoin core or Electrum gives you exclusive personal control over some funds. This is what is called not imprisoned Wallets – There is no third party custodian that takes charge of your money.
Coinbase accounts are guard. As far as the Bitcoin network is concerned, Coinbase has money in coinbase accounts, not you.
Ownership of funds in the Bitcoin network is determined by who holds the secret numbers known as private keys. Money is tied to a private key. If the wallet doesn’t have the private key for some money, that wallet won’t have access to that money to spend.
If it helps, imagine having bank accounts in two different banks. Your bank statement from one bank will not include any money in the other bank. You cannot direct any bank to send money to someone in the other bank. Bitcoin does not work like banks but has a separation of control where there are many private keys in use.
How do I recover wallet funds?
You can, for example, have your dishonest wallet display an address to “receive bitcoin”. Then log in to coinbase and use its facilities to pay money (or withdraw money to) this address.