As snotty as this customer sounds, from what you've described he's in the right, as annoying as that may be.
As for the T's & C's, it's no good pointing people in their general direction. Unless you have their signature on a paper copy of those T's & C's, they haven't agreed to diddly squat.
For future work, it might be best to state at the outset that you want staged payments, and to include it in your T's & C's which you should be requesting that in order to carry out the work, you would need the customer to read, sign and return (self addressed envelope of course) prior to engagement.
We do this for all work over £1000.
It's a$$ covering mate, that's all