If you're speaking about "testnet4", you can easily setup your own node within an hour since its blockchain isn't that big and most blocks aren't full. Given that you have a good hardware.
That's given that you have a good hardware.
Just start bitcoind and bitcoin-cli with --testnet4 arg when you start your node and issue commands.

The issue of public RPC nodes is they aren't willing to allow the public to use those types of commands (submitblock in your case) since it can ruin their node's ban score.
So most of them just blacklists such command.

Example, those commands are blacklisted in the public mainnet demo of this repo: github.com/janoside/btc-rpc-explorer
That repo's live demos used to have a working testnet and signet versions but for some reason, both are not accessible but AFAIK those are also blacklisted there.

If you're speaking about "testnet4", you can easily setup your own node within an hour since its blockchain isn't that big. Given that you have a good hardware.
Just start bitcoind and bitcoin-cli with --testnet4 arg when you start your node and issue commands.

The issue of public RPC nodes is they aren't willing to allow the public to use those types of commands (submitblock in your case) since it can ruin their node's ban score.
So most of them just blacklists such command.

Example, those commands are blacklisted in the public mainnet demo of this repo: github.com/janoside/btc-rpc-explorer
That repo's live demos used to have a working testnet and signet versions but for some reason, both are not accessible but AFAIK those are also blacklisted there.