I think you could add Jokers and/or FSL if you're smart. If you're adding them to everything then you might decrease the extra squat work back to 3x5-10 with 40, 50, 60% like the original template.
If you're following the template as written then I don't think you'll have any problem. The only real extra pressing work is on day 1. Overall I think the total body program has much less volume than any other template. It has no dedicated assistance programs like BBB.
Play with your programming and see how you feel after one or two cycles. I think the total body template is about the only one you could do this without screwing things up.
One thing I've been considering is switching to a more traditional 5/3/1 template and just running it 3 days per week so I can do some of the challenges. It'll take longer to run a cycle but I don't think that'll matter too much. I doubt any of us will be able to continue adding 5-10 pounds to our TM for too long so taking longer to run a cycle will just mean you go longer before stalling and resetting.
In regards to assistance work just so what you feel you need. Work on weaknesses. I superset pull ups with 5/3/1 warmup sets every day. I also throw in a horizontal pull movement like face pulls, reverse fly's, or rows each day. That's more for personal reasons; I'm paranoid about having an under-developed back.