I have the same as you - TNT Sports as an add-on via EE, and HBO Max free with Sky. I have logged into the app using the same email address (my email address I use for my EE account is same as Sky). The email address on HBO Max automatically links to the EE account with TNT Sports, however if you activate HBO Max on Sky, both of your accounts are combined. So there is no need to "flick between two accounts" as they are both combined together in the account on the app once activated.
It just gives you a warning when activating via Sky, that you may be billed from two accounts, which is fine as it is picking up the fact that you have accounts with both EE (TNT Sports) and Sky (free with ads). What I have noticed is I appear to be getting UHD on movies, so it looks like Sky gives me the access to movies etc, and EE gives me access to TNT Sports, and also makes everything UHD
