Hey there, welcome to the forum!

I don't want to mess around with LiPo's as there's no need seen as I'm not actually fighting with it - what other batteries are viable?
I use LiFePO4 batteries in my Beetleweight Shu! which worked quite nicely at the last Beetleweight event I was in though they give out less power than LiPos (a 3S LiFe battery is 9.9v nominal, 3S LiPo is 11.1v nominal). The other alternative is NiMH batteries but the Lithium batteries are most prevalent.

I need help deciding on what ESC, servos for lifter, transmitter and batteries. Something that is suitable for what I'm doing for also has potential should I want to use them in a proper fighting bot!
Get a transmitter with mixing built in (like the OrangeRX system) as that will cover you in the event that your ESC doesn't do the mixing for you. As far as ESC's go, for a simple plug and go affair, I use a Sabretooth Dual 5A board. A bit expensive for what it is but it works. I know some folks here have had mixed results in the past but I've not really had any issues in using one.

With Shu! also being a lifter, I don't use a servo, I use a single Pololu 37mm 67:1 motor (previously a 131:1 motor but I broke the gearbox) which again, gives enough oomph.The 131:1 motor was enough to lift both Shu! and the opponent before it broke.

I'd recommend going around the build diaries as well to see what works and what doesn't.