The word "budget" sticks out in the original post, so I'm going to recommend the TW add-on.
Taildraggers available for solo rental are very hard to find these days, but there are plenty available for dual only, and they are all usually cheaper to rent than any given HP machine.
My own TW experience illustrates this nicely (and the Champ in question is still available, for training and solo rental, down at KSDM):
-$65/hr wet now, for a well-maintained '46 Champ. Find me a 200+-hp airplane for rent less than $100/hr wet...!
-Ten hours were required to meet their insurance requirements for solo flight (insurance is what's gutted the old taildragger solo rental market), but it took me exactly 11 hours of dual to get the TW sign-off anyway. Instructor said that was about average for transitioning pilots.
- The instructor was not cheap; $40/hr, but he was the perfect guy to introduce me to flying antique taildraggers, being something of an antique taildragger himself. He'd been giving dual in that same airplane for over 40 years!! Also had flown Trackers off Navy carriers during the Vietnam years, so his overall aviation knowledge is complete, and he does not scare easily.
With all that experience in the Champ, he is an expert at figuring out what you've got, then systematically turning you into a tailwheel pilot. He was worth every penny, IMHO. I did better than I though I would, and still vividly remember all the important things he told me. I went on to log another 15 or so hours on my own in that bird, and did just fine with it. It has a comm radio but no tx, which wasn't limiting to me- I flew it all over that area, from San Diego to Chino, through the VFR corridor, out over the hills, over the beach... most of it at 80 mph with my elbow on the left windowsill and a chart in my lap.
Champs, BTW, are very good beginner TW airplanes- flown solo from the front, good vis over the nose in the 3-point attitude, docile and predictable, but with a challenging mix of much adverse yaw and not much power, it provides an "authentic" experience. And like any taildragger, the center of mass is behind the mains, so it will keep your feet busy.
Getting the endorsement served no "practical" purpose, although I was planning, and still am planning, or shall we say hoping, to get a nice antique 2-seater of my own someday. Over the next couple of years my interest, enhanced by the endorsement, got me stick time, and takeoffs and landings, in other classic TW airplanes, courtesy of very generous owners who wanted to encourage me.
I have to agree that the TW skills don't automatically make you better with nosedraggers, so it's not that, either... I just like antique airplanes, and the challenge of transitioning from "Land-o-Matic" gear to the "when men were men" gear configuration was a fun shot in the arm for the "another rented Skyhawk" blahs.
I'd love to get every possible add-on rating; most of us would, but if you feel the need for something new that won't break the bank, you should consider taildraggers.
Or gliders...
