I think there is a confusion here. Logs provided by mediasoup-client are just those: logs produced by mediasoup-client. They are not intended for the application to inspect them. Your app can produce logs or send them to the server or whatever.
I strongly think that what you are asking for just satisfies your specific app needs. IMHO, your app should be interested in collecting stats and so on, and transmitting them to your server if you wish to inspect/store them. The logs produced by a client side library are just that, internal logs that print some information, nothing else.