Upgrading towers after defeating Father would defeat the purpose of the challenge :) So you spend whatever money you have after Father is defeated, and after that to only move towers, but not to buy / upgrade them.
The idea there is to use clovers to clone EMP, and sell them, which gives you extra 3600 to spend.
Another question is whether it's intended that when the summoner is defeated, the game doesn't award money for the remaining spawns. This way, if I have say mostly Ice / Teslas by the time I fight the summoner, most of the spawns would die from the shards / lightning chain, and I'd end up with more money than if I mostly have stones, that kill the summoner before most of the final set of spawns. I made some experiments, and the difference in money made from the summoner wave can be as high as 100+ extra gold depending on how many spaws remain alive when the Summoner is killed.