sslkeylog | Log SSL/TLS keys for decrypting SSL | TLS library

I would like to decode and store raw SSL/TLS data in order to prove something in the future, i think it will be better if i describe it by an example:

1) There is a remote REST endpoint that is accessible via SSL and provides some useful data. This endpoint is controlled by some third party and i have no control over it.

2) I issue a GET request to this endpoint and store the raw SSL traffic of the response in one file and then the decoded ssl traffic in another file

3) Additionally i save the SSL certificate of this endpoint in a separate file (i guess this operation would only need to be done once)

If i understand correctly when i have all the data from the steps above and in the future, when another third party asks me for a proof that there was certain data on the remote server at some point in the past i could:

1) Present them with the certificate i saved (step 3). Since this certificate is signed by a well known authority there is almost zero chance that i could forge it

2) Present them with the RAW SSL data along with a way to decode it using the certificate above

3) In the decoded REST response headers remote server time should also be present.

Therefore if i am not mistaken i would be able to prove, with almost no doubt, that at some point in the past there was that certain piece of data on the remote party's server even if the remote server itself is no longer operational.

Therefore i can use raw SSL data to prove that something existed on the remote server in past. And there is very little chance of someone challeging this claim because forging the response data or headers would mean that i was able to break the SSL encryption.

Is there a way to automate such scenarios? I mean is there a way to record raw SSL data in a form that would allow it to be easily decoded later?

I have tried using firefox/chrome browser along with the sslkeylog files and sniffing the traffic using wireshark - it actually works, but is there a way to automate it using some kind of REST library since using firefox and recording SSL keys each time can be troublesome - there may be a need to store thousands of these requests per hour and doing it manually would be a full time job, additionally i am not sure if there is a way to prove that these keys stored in ssskeylog files actually match the certificate of the remote server...