Experts and BHG readers answer.
Invasive grass and plant
A number of the things you've tried should do the job. No offense, but I think the problem isn't what you're doing, it's how you're doing it. Bahia is a tough week, but it's not invincible.
Roundup is the most effective weed control, bar none. Aggressive weeds with deep roots often are able to grow back and need a followup spray when green shoots start to reappear, but two (three, rarely) sprays are generally enough to wipe out just about any weed. Once eradicated, a weed fabric and mulch is big help. Just remember to spray the edges once in a while to prevent the bahia from creeping in from the edge.
One other thing you might consider is a grass killer. There are some chemicals available that selectively kill grasses without harming ornamentals. Grasses start to intermingle with ornamentals and they're nigh impossible to remove, and they keep re-invading the rest of the flowerbed from there. A grass killer will fix that. BE SURE YOU HAVE THE RIGHT STUFF. It must say that it's a selective grass killer. The most common one is fluazifop (that's a chemical name, not a brand name). Sometimes Roundup will go by a name like "weed and grass killer" and that would be a bad thing to spray on your ornamentals.
For the petunias, Roundup still can work, but sounds like they may be germinating from seed too. Try a weed preventer like Preen. Heavy mulch will help too, and weed fabric.