Can't be done. Period.
*IF* you have control of BOTH servers, you can use message events to pass data. That is, you can write special code for the uk1.roomlynx.net site that would accept messages sent from the www.lakedistricthotels.net
site. But if you can't alter the code on uk1.roomlynx.net site, then you can't do it.
You *MAY* be able to do some of this with the help of server-side code, but it depends on how carefully controlled that roomlynx.net site is. Since it is https:// you may find it quite difficult. Likely the best thing you can do is contact the owners of roomlynx.net and ask if they can provide you with an API for this purpose.