I'm not sure if you have a two speaker system or a 4 speaker system(not including subwoofer). If you have a four speaker system, then it wouldn't make sense to provide more power only to two speakers.
If you have a two speaker system, then it would make more sense.
From what I am reading, you have a soundbar, and therefore a four speaker system.
So, NO, I don't think that amp is a good idea. It maybe a cheap solution, but not optimal. For your setup, a 5-Channel Amp, would be the best solution.
This way you power the speakers using four channels, and the sub has its own, more power mono channel.
From what you are saying, the shop intends to Bridge two channels for the subwoofer, and provide additional power to only two speakers. It might work, and give you a little more power, but it is not optimal. I guess you will keep your fader capability this way, but I don't like it.
The discussion link below provides information and Crutchfield discussion links.
Can i hook up 1 sub to a 4 channel amp? - Yahoo Answers
Whatever you plan to do, ultimately, you'll want to upgrade your subwoofer. That stock paper subwoofer will not suffice for increased power. You can get a 400wattRMS subwoofer for $99(Kicker 40CWR84).
But, if you do go with the audio shop recommendation of a 4 channel amp, you'll need to know the ohms that they are going to wire to the subwoofer. Then, get a upgraded 8" subwoofer. You can get 200watt-400watt subwoofers, from 1-4 ohms, for $49 to $99 dollars. The MTX, Kickers or JL seem the best. The Rockford Fosgates look way too deep in depth to mount in the stock enclosure.
Other Wrangler discussion are saying that the 8" 400watt Subwoofer will shake the windows, if properly powered. This is why I say you should get a 5-Channel amp, which will properly power the sub and the four speakers. The upgraded subwoofer will need a spacer as well, because the powerful magnet needs more depth than available with the stock enclosure.
There are many discussions about it. I've pasted some of the discussions below. The stock rear enclosure seems basically the same from 2007 up. I'm interested in what you ultimately decide to do.
Aftermarket Subwoofer in Factory Subwoofer box anyone? - JeepForum.com
Kicker CRV8 in factory subwoofer enclosure install.
direct subwoofer replacement
What would I do? If I had $700 dollars, I would get an Alpine PDX-V9 or another 5-channel amp, and get the Kicker 4ohm 400watt subwoofer. Then save your money for better speakers later. You will get all the thump you need, and the Alpine is ready for your speakers later. Maybe the Shop can recommend a cheaper 5-channel amp that provides at least 400watts to the subwoofer, just in case, the shop wants more money available to do the install. Hope this helps. Good Luck.