To me, part of working a frog is, for lack of better terms, wiggling the hell out of it as I retrieve. A heavy, long rod makes this difficult and wears out your hands and forearms. I like a 7 foot or nearly that heavy rod, I use either a 7 ft IM7 Gander Mtn rod ( two hand handle) or a 6'9" St Croix that is designed for topwater ( both are bait casters, which is my favorite. I love the St Croix in terms of being able to work the lure, but sometimes it is a bit soft for hook sets. The other rod is a bit bulkier and harder to work, but works well enough and I think I can get a harder hook set. I also have started morphing to a different hookset. Rather than wail away with the rod, I lift it a bit and reel like hell, loading the rod. I seem to get more hooksets this way and when I miss, the frog doesn't leave the county.
I LOVE working frogs, especially down south, and do it a lot.
My frog rod is a Gander Tournament 7' heavy. Its light, cheap and has a warranty. I think they are $80 but routinely on sale for less. Actually all of my kayak rods are the same money. They're quality rods but also inexpensive enough that if I lose one over the side, I wont lose too much sleep.
I use a Kistler Mag TS Jig' Toad Special and while that rod is out of budget and not made anymore, you might be able to find one on ebay. I will say that the benefit of a higher end sensitive rod is not always the ability to catch more fish.
Lighter weight rods nicer to fish with because you get less fatigue.
More sensitive rods will be funner to catch fish on because you feel more of the fish.
Sometimes you find another good application for a rod other than what you intended it for that may need sensitivity. There are always new tactics coming out.
I am a big proponent on paying as much as you can in a rod for any application, but if you are going to budget on one application...a frog rod might be top on that list because you are using heavy braid and more than 50% of the time you will be pulling in more weight in slop/weeds than actual fish!