I already posted my thoughts on this on the Discord server, but I'll put them again here.
I own both instruments. The mellophonium can have musical value, while the frumpet kind of can't. I bought the mellophonium because I really wanted one (and it was dirt cheap); I bought the frumpet because it was dirt cheap. That pretty much tells you the whole story, but here's a more thorough comparison.
1. The mellophonium and frumpet both have a unique sound. The mellophonium's is beautiful
in the right hands. The frumpet's...is not. It's very forgettable and a tad annoying. Kind of sounds like it has a cold. It can be a good blending instrument if you work at it, but that's about it. If you just want a unique alto brass instrument to be a blending ensemble instrument, buy a DEG model 1220 alto cornet. They also show up on eBay pretty frequently and play very well (since they were made by Willson). Nice sound, very compact, sturdy, fun to play. I've used both mellophonium and alto cornet extensively on both live gigs and studio recordings...the frumpet sits stored in its case on the other side of the country.
2. The Conn 16E is NOT a well-designed instrument. Its woeful reputation IS overblown (largely due to the early Kenton section detesting having to play the instrument and deliberately playing it as disruptively as possible), but it is far from an easy instrument to play. The valve slides are too long for F and too short for Eb and the high range is whiny and very difficult to center. Having said all this, the frumpet is
FAR worse. Partials are no more than suggestions, and the ones that exist enough to kind of hit are hilariously out of whack. I've never played an instrument that had to use more alternate fingerings just to get anywhere close to in tune.
3. Both instruments were designed as marching substitutes for the horn and didn't really do a very good job (hence the modern marching mellophone), but the mellophonium has an established history in jazz, both as a solo instrument (especially if you include
what Don Elliott did with his custom precursor mellophonium) and an ensemble instrument (in the 4-person Kenton section). Stan Kenton chose mellophoniums as his 4th horn section very carefully...he tried other things including horns, flugelhorns, and I think alto trumpets if I remember correctly, but chose the mellophonium. The frumpet has no such illustrious history...it failed as a marching instrument and failed at everything else. Contrary to popular belief, the Conn 16E was far from the only model of mellophonium...Holton made 2 different models (!), and Reynolds, Olds, and even Courtois made them. Getzen was the only one to make the frumpet, and there is a reason for that.
4. The mellophonium looks way cooler than the frumpet. However, the frumpet is much less of an ergonomic challenge, being a familiar trumpet shape with a familiar trumpet grip. That said, the mellophonium is easily manageable if you know how to hold it right. (Pro tip: grip the 3rd valve slide with your left hand, and angle the instrument to the right a fair bit. Works just fine.)
Now moving on to the two specific instruments for sale.
$125 is a lot for a frumpet. I got mine for $100 in good shape with both slides and a mouthpiece, and to be honest that's the max I would ever pay for one. $200 is steep for a 16E as well, but the complete package of good condition, Eb slide, and original mouthpiece is pretty rare so I would happily pay that money. Mine was $110, but it came with no mouthpiece or Eb slide and needed leadpipe repair.
In short:
Which one of these choices is less idiotic?
The mellophonium. All day, every day, every way, and all 6 degrees from Kevin Bacon.
Neither of them are "idiotic" if you enjoy obscure brass instruments and just want to have fun, but the mellophonium is the more relatively sane purchase. Frumpets show up for nothing all the time, good condition 16Es with all the original extras are not that common. If you really decide you want to get a frumpet, you can pick one up later on for around $100 and have both. But the mellophonium (especially that specific 16E) is a much better first buy.
IMO, of course.
P.S. If you want more obscure instruments to buy on eBay for pocket change, add that DEG alto cornet (usually/always listed as a "DEG mellophone", but they never made a mellophone so it'll for sure be the right instrument), circular mellophone (especially if it has multiple crooks, usually F/Eb/D/C), and flugabone. You can also find older marching mellophones and marching French horns for cheap, but those aren't as unique as the others.
P.P.S. There is one BIG caveat to the frumpet. IF you decide you're ok with sinking more money into the instrument, the frumpet can be easily modified into a playable instrument. The easiest route is buying one of those Chansons hybrid pieces with the French horn shank, thus eliminating the single thing that makes the frumpet so awfully out of tune. (The horn mouthpiece and its deep V cup is not an acoustic match for the instrument.) Level 2 is to swap the leadpipe for an alto horn shank and use an alto horn mouthpiece with it. Level 3 is to swap the bell with a small trombone bell and the leadpipe with a trombone leadpipe, thus turning it into a nice alto valve trombone. A member of the old TTF used to do those conversions for people.
BUT, if you want to spend more hundreds on the frumpet (a ridiculous concept), you might as well just buy both the mellophonium and the frumpet.