I have my eyes on a 2018 330e.
But, it would take about 3 months to get the approval from my HOA to install the charger in the parking garage.
Would it be detrimental to this car to drive it strictly on the gas engine for that amount of time?
No, this vehicle as well as similar "hybrid" vehicles are designed to work this way. The gasoline-fueled part of the system will keep the batteries properly charged.
This vehicle only has a very limited electric-only range (30-40 miles is what the ads say) and so unless your trips are always under this limit, the vehicle is going to be running on gasoline most of the time anyway.