Is this a safe way to prove the knowledge of an ECDSA Signature?
Any EU-CMA-secure signature is indeed a proof of knowledge of either a signature of this document or the private key.
So, what you have done is using of second ECDSA signature as a proof of knowledge of discrete logarithm $s$. This completely makes sense, and I don't see any attacks on this.
Between, as a proof of $s$ you can use also Schnorr signature or Schnorr NIZKP - it has a property of zero-knowledge in random oracle model (while DSA has no such property).
Also, this idea of chaining signatures in this way is used in identity-based Galindo-Garcia signature, where 2 Schnorr signatures are chained.

But I can't answer you completely with a proof of security.
 At first, we need a rigorous formal model of attacker.
This cryptographic primitive (proof of knowledge of signature) is not very widespread and seldom considered.
Some works are devoted to this problem though, you can look here: https://link.springer.com/chapter/10.1007/978-3-540-47942-0_9# .
 Aldo, DSA signature has poor provable security in general (compare to Schnorr one).
E.g., if you used Schnorr signatures in this way - I would exactly say that it's secure (because Galinda-Garcia identity-based scheme is provably secure).
And, this solution could be safe on practice, but it lacks ZKP property for sure, so researchers mostly devote their efforts on other constructions, which provide ZKP.

Don't consider my answer as complete and accurate, this is just general abstract thoughts. I hope it will help you a bit.