You can put this client into a different Client Group, where you change the number of attempts. This way, she can have her card attempted every day for more days in a row, and everyone else won't be affected in the same way. There's no way to attempt and not send out an email regarding the attempt though.
I agree with @Blesta Addons you should update and have the custom module updated. In saying that I would think that your module would be written to connect via http or https (not on a TLS protocol itself) and it would be down to how your server is setup as to what version TLS is used, more to the point their server since your connecting to them. I would think as long as your system/server is up to date then it should be able to communicate on all newer TLS protocols.