Deploying via IPFS might work indeed, but you're then either reliant on a gateway or have to run IPFS yourself locally. I know that Brave now supports IPFS, but for most people a more lightweight solution would be to install the Signed Pages extension for Chrome or Firefox.[0]

I'm not sure if you actually need a trustworthy backend, or rather, if the frontend is encrypting its state before sending it to the backend, then the worst a malicious backend can do is delete or replay your data. By including timestamps and replicating your state to multiple independent backends, that risk can be largely mitigated.

[0] https://github.com/tasn/webext-signed-pages