Further North

What rod do you have?
 
I ask because you're going to want to try to balance the rod with the reel.

An inexpensive, heavy reel, like the Reddington Behemoth, will make a lighter rod/reel combo butt heavy and not a lot of fun to fish.

On the flip side of that, a reel that's too light will make the rod/reel combo tip heavy, which is arguably worse.

We also need your definition of "break the bank".  Fly reels span the gamut of prices, from cheap reels under $100 to $1,000 or more.  Going used can make that even more confusing.

Share what you've seen so far, and what you're going to be comfortable paying, that'll help.

You also have a decision to make about the line, and how you want to fish; and knowing what rod you have will help figure out what grain weight line to use (it's not as simple as just getting a 7 wt. line to get the best fit.)  

Are you going to want to fish topwater poppers?  Streamers subsurface?

I'm not sure where you're fishing, but where I live, a 6 wt. is the minimum size for decent bass flies, a 7 wt. is mid-range, and an 8 wt. is the all-around most useful.  I'm not trying to talk you into another rod, but be aware the the 7 wt. is going to have some limitations on size, weight, and wind resistance of the flies you'll be able to cast well. 

There's a "thing" in fly fishing where anglers like to use a rod that's not enough rod for the task...do yourself a favor and avoid that...it's a short path to frustration and lost fish...and ignore experienced fly anglers who tell you "All you need is a 5 wt.!  I caught a 5 lb. smallie on a woolly bugger on one."  Yeah, an experienced fly angler can do that.  Sometimes.  They don't tell you how many fish they lost along the way.

Off topic, but important: You're going to want to pinch your barbs.  Not for the fish, but for you and whoever you're fishing with.  You will stick yourself with a fly, it's not a matter of if, it's a matter of when...there are times when you'll have 50 or 60 ft. of line out and on a cast...and gust of wind will push the fly into you.  A pinched barb mean it takes five seconds to get back to fishing.  Big hook past the barb can mean the end of the day on the water and maybe a trip to the ER. 

You're likely to hear that for fly fishing, all the reel does is hold line - this is true to some extent, because you'll never need to put a bass on the reel - you should always strip them in.  If a bass puts itself on the reel (I've never had this happen) that's fine, but never waste time and focus trying to put slack line back on reel when you have a fish on...that's you you lose a tight line, and lose fish...

...getting to specific reels: Lamson is a great choice, and relatively inexpensive, even new.  Their Liquid S is about $170 in the 7+ size you want and their Remix S is about $280 in that same size.  You can go up from there.

Great drags, light weight (often the lightest for any given size), durable, and very well made.

If you go used, you'll see lots of options, anything that's not beat up from a recognizable brand will probably be fine - but stay away from heavy reels.

That's probably way more than you expected, or wanted, but I've found that the way to get fly anglers to have fun and come back to the sport is to set them up for success.

I'm more than happy to answer more questions, fire away.

Another follow up:
 
We were met at the put in at another river by some young men - roughly the age of the river-owners in the original post.
 
They arrived a few minutes after us, and they parked, walked over as we were launching, introduced themselves, and asked what our plan was for the day.
 
When we told them we'd be floating down-river several miles, they told us they were planning to work the area around the dam, then head for another river for a short float.
 
It's things like this that inspire me to think we're not doomed.

MK has shared that 1540 is the top end for their motors, so
 

In the 4th picture down, there's a foot long pink blur in front of me in the rower's seat.

...that's the fly.

We're working an outside bend hole that's maybe 10 or 12 ft. wide, and 50 ft. long.  That's where the big girls hide.

Thanks!  On a fly too!
 
...I don't wander into the bass area often...