How critical is TA experience in grad school if I want to pursue a career in academia? I got accepted into my dream program at one of the best universities in the US, and I'll be working under an excellent, very well known advisor whose interests align very well with mine. However, since I will likely be getting a multi-year fellowship, and due to how this specific program is structured, it seems that I won't be getting much TA experience, if any at all.

I love teaching and enjoy it very much, and I already have 2 years of TA experience as an undergrad. How much would it affect my future TT applications if I have a (hopefully) solid research record but not that much TA experience? I don't want to turn down this opportunity but might have to if it turns out I won't get the TA experience I want/need.

EDIT: My field is applied physics/electrical engineering/materials science (very interdisciplinary research area so in theory I could be hired by any of these departments). I hope to work at an R1 school.
Two years is probably enough, especially since your two years' of TAing went so well.  You may want to concentrate on demonstrating yourself as a researcher now.  However, if you have time to spare, I'm sure you could offer your services to the person in your department who coordinates TAs.  As I probably don't need to tell you, you'd want to keep your advisor in the loop.

Consider that a fellow or research assistant is sometimes pressed into service to lecture in place of the advisor in case s/he has to miss one or more classes.

When you feel ready, you could expand on that idea, and ask someone (probably your advisor, but it could be someone else too) if you could be a guest lecturer for a particular topic in a particular class.