I won't get nit picky about the visitation issue... but it is not related to the child support issue. Unless you have joint custody and each have the child the same amount of time (vs. the child having primary residence with one parent and "visiting" the other parent on weekends or whatever the schedule is), visitation actually has NOTHING to do with support.
It also has nothing to do with him saving money to move out of his parent's house.
Support is determined by a formula (called "guidelines") that only needs a couple of things to spit out the support number. Your incomes and how many kids. Actually, you don't really even need both the incomes... just the one who is paying support is really all you need. They plug in his income to the guideline chart and it says what his support obligation for __ # of kids should be. That does NOT include insurance or other things he should be providing, but if he does provide insurance, then that can be deducted from his gross before running the guidelines.
The child support office may very well only review cases every 3 years. That isn't surprising. Do you have any idea how many cases are routed through those offices monthly? But their intra-office rules about how often they review cases has NOTHING to do with how often YOU can file for a modification through the court. It's just how often a government agency has determined is reasonable for them (the child support office) to review it internally. You can go retain an attorney to file for modification tomorrow if you desire. The child support agency is NOT the court system. Please don't be confused about that. They may act that way sometimes, and they may actually believe it themselves sometimes... but they are just bureaucrats. Not part of the judiciary and most likely not even lawyers... they have to contract the actual LAW work out to an attorney.
I worked for an attorney for 10 years, who prosecuted cases for a state child support agency. Trust me. You are not required to use the state child support agency to file in the court system. You are probably getting a "sweet deal" by doing so in terms of court costs and attorney's fees. It may not seem that way until you check out your options. But you are free to do that at any time you want. So it is really up to you how you want to proceed. Go back to the CSE office and ask for a supervisor and follow up trying to get the 'system' to make an exception to their "rule", (or show them how their rule doesn't apply to your situation?), or go consult with private counsel. Or both... and then decide what it is in your (and your child's) best interests.
In the eyes of the court, child support is far less subjective than you might think.