You need to be careful using that moniker because it can be taken very negatively. UH is on the rise and does not deserve to be referred to as a high school from an academic standpoint.
In regards to it being a commuter school, UH actually has a higher percentage of international and out-of-state students than UT. And if the bulk of in-state students do live at home and save money by doing so, more power to them.
The term "Cougar High" makes it sound like the school has terrible academics. At least that's what it sounds like to me. I'm not suggesting that this is what anyone here means when they refer to it as such, but that's the perception I'm left with. There are very smart and capable people that go to school there and have graduated from there.
I would also bear in mind that it is getting increasingly difficult to get into UT. Would you rather have people you care for that aren't necessarily your children go to A&M or UH as their alternate school after not getting into UT? If you degrade UH academics, like many ags I know certainly do, they'll be less likely to choose UH. I see the school as a quality option for those unwilling or unable to get into UT due to cost, location, academic qualification, etc.
If it is a term used only in reference to their sports programs, is it really necessary?