You are mowing too low. Raise your mower to the highest setting and extend your mowing to once every 14 (or 21) days. That will shade out the bermuda and you'll have a dense St Aug lawn again. There is never any reason to lower your mower with St Augustine. I tell people to weld it in place so your neighbor kid doesn't make the mistake of scalping it when you're on vacation.

How are yo watering (frequency and duration)?
What are you fertilizing with and how often?

To your question, yes, let the bermuda spread. Its a softer grass, will require the same fertilizers and amount of chemicals to have it look good, it holds up to traffic better. Also, your not going to win if you want to remove the bermuda, its there to stay. You will have a much larger selection of chemicals to put on the lawn too, with bermuda.

Beg to differ. I have large areas of my San Antonio lawn which were 100% bermuda in 2011 which are now 100% St Aug. I've done this repeatedly since 1992 in various parts of the yard. For the first 10 years I cycled my natural driveway between common bermuda and St Aug. Withhold water and mow short and you'll get bermuda. Water plenty and mow tall and you'll have St Aug. 2011 was a bad year in San Antonio and my back yard died. Bermuda from the side yard filled in at the end of the year. In 2012 I nursed the St Augustine and slowly recaptured the yard. Now you'd never know it had any bermuda.

If you're doing St Augustine correctly, you won't need any chemicals, so having just a few isn't a problem.

According to the Bermuda Bible, bermuda needs to following care to remain dense and great looking.

Weekly
Mulch mow 2x at 1-1.5 inches
Water 1x

Monthly
Fertilize heavily with a high N fertilizer

St Aug does not need twice weekly mowing, in fact I go the other direction with mowing every 14 days. If yo want to fertilize monthly you may but only with an organic fertilizer. Fertilizing in the summer with a chemical fertilizer often leads to dead grass.

A few things I will disagree with you on your statements. A cinch bug outbreak and left untreated would kill off a St Aug lawn. So preventive maintenance is required.
Bermuda grass, as I see in my neighbors St Aug lawn, grows side by side at 5" tall, with the St Aug.
As for using non-organic fert in the summer on bermuda, the grass grows thick and lush. Im not talking about urea based ferts but nitrate ones.

I would like St Aug if two factors were breed out of the grass. The wideness & stiffness of the blades and the inability to handle heavy traffic.
I like the softness of my zoysia bermuda lawn and my dogs can not wear a track in the yard.

To each his own.

Your experience with zoysia is different from mine. Different zoysias for sure. My Shadow Turf is definitely more compact where we walk on it.

We get chinch bug outbreaks every year and people do not lose entire lawns. Preventive applications of insecticide would be unnecessarily damaging here. Preventive applications of something like beneficial nematodes might be useful, but you have to get that application just right or you wasted your money.

I have a second yard in George West, Texas. I am watching a spot with a mix of bermuda and St Augustine. So far the bermuda is happily intermingled and growing at 12 inches high. I am seriously in denial at this observation, but I do have to face it. That is definitely not the situation in San Antonio. Shade is much different in San Antonio with no morning sun at all in the areas where I have been transitioning back and forth from bermuda to St Aug.

Bermuda lawn is definitely more work compared to St Augustine from my experiences. I'm not a big fan of bermuda. Too picky for my taste.

I've never done anything to deal with cinch bugs so I'm not sure where that came from. I don't think it's a big deal. At least when I had St augustine for 7 years with no problem. Minimal fertilizer - just 2-3 times a year, yearly fire ant killer. That's it. Bermuda.... never ending weed problems... brown spots from not mowing enough (once a week is not even enough to prevent that). St augustine is much more forgiving.

If anything, i'd grow 'Palisades' zoysia over both. Just very expensive. They win with very minimal input from fertilizer, etc.