Is it possible to claim clams by signing a message with your privkey?
Yes, but the message has to be in a particular format. It has to be in the same format as a regular transaction. For the inputs, list the CLAM utxo's you're spending. For the outputs, list the address and amount you are claiming. Then sign the whole thing using your Bitcoin private key.
There is a reference implementation that can do this for you in the CLAM github - it's part of the CLAM wallet - but you are free to rewrite it if you like.
Giving the privkey to some website, or some other software (even if it might be legit) is not a valid option.
How are you going to sign a message if you're not willing to use a website or some software?
I'm aware that you can first empty the BTC address to a new one, but that's not an option in some cases (like, giving up my sig address).
Maybe this is the part you're not understanding. What do you mean about "giving up my sig address"? Do you think that when you sign a message with an address you aren't making the address known? Because you are. Every time you publish a signature you are publishing the public key along with it.
If signed proof isn't possible/planned, then I'll just ignore my clams to be on the safe side.
Signed proof is how every transaction works in every crypto. If you're looking for a way of signing a message without revealing the address used to do the signing I don't think you're going to find it. If you don't want to trust the free open source code provided by the CLAM project and used by thousands for years without claims of a single theft resulting from its use, you are free to audit the code or even rewrite it from scratch. And of course you're free to never claim your free CLAMs too.