Plumbing – Why does the shower head stop abruptly after a few minutes

The stopper you're describing is called the diverter. The symptoms you're describing sound like it's gotten slightly bent or nicked. I would take it apart and try re-setting the pin. Maybe scrub it a bit with a light abrasive to true it up. If that doesn't work, replace it. I assume your diverter is in your tub spout. You can usually remove that by removing the hex-head set-screw and then rotating the entire spout (the padded handle of a hammer stuck in the spout can help you get a grip).

There are a few ways you can improve the situation but there isn't a true fix.

1. Replace/fix your shower valve. The rises and drops in your water pressure due to flushing the toilet or night time (city water pressure rises due to low demand) is allowing minuscule amounts of pressure to seep past the valve, with large amounts of time in between this could be air.

2. Redo all you connections with new Teflon tape, since this came about with the new faucet you may have a connection that is allowing air to seep in and either allowing the heavier water out or just assisting in the former scenario.

3. Install a backflow prevention valve or check valve to prevent any backflow, it sounds odd since water to coming out but somewhere it's exchanging water with air and this can help prevent that.

4. I can find the product name, if someone knows it please edit or comment, but there is something similar to a vacuum relief valve that you can get. This would be the most outwardly visible solution however it should fix it. The item looks like a vacuum relief valve but works a bit differently; it's a valve that is 3-way valve that has a pass through, your main water flow, and a vent. The vent is open when there isn't any pressure, or very low, to allow any small pressure build ups after the shower to simply vent and it would close with pressure so it wouldn't spray water during your shower. This should be installed at the highest point of the water, right before your faucet should be fine, and pointed up as it should only vent air. After you shut your shower off this would open and allow you to drain the line, just lower the head/wand; now if you have any water leak past the valve it would have to fill the line before over flowing, a ball valve one wouldn't even overflow. The item is either sprung or is a weak ball check valve.