you did not mention which browser you are using.
I hope it is IE. If yes, maybe you can stop your computer's anti virus software and install plug-in again to see if it works.
(1) Does Hik-Connect play nice with the LTS-8704?
Not sure. But most LTS device are made by Hikivision.
(2) Does the LTS-8704 support Hikvision's "Cloud P2P" which apparently does not require a fixed IP?
yes
(3) Any other ideas to meet client's requirement without paying for a fixed IP...