I agree with the others:
[1] No, you cannot prove ownership of a certain amount of BTC like you can on L1.
[2] I wouldn't recommend anyone to send channel state commitments to anyone else than their channel partner.
What if the "commitment transactions" information was sent to me only (assuming I am honest and trustworthy). Do you think this is the best way to proof ownership of funds in a lightning channel?
That's a pretty big assumption that could cost channel partners their whole balance if you were to (mistakenly or not - Bitcoin don't care, and it's irreversible, immutable, right..) publish one of those commitments.
Also do keep in mind it will be a pretty annoying
TOCTOU problem. Sure, even on L1, you could check someone's balance and just 10 minutes later it could be different. However in Lightning, this can change hundreds of times per minute or maybe even per second. I could lend 1BTC from someone through Lightning, make a snapshot of my channel states, immediately return the funds and send you the (now outdated) channel state, making you believe I own 1BTC while I don't.
There's just no way to check if the channel state was updated, since this information is private
by design.