What's the U bend for? To locally speed up the flow?
Most consumer type ultrasonic flow meters are settling around an acoustic path that goes from a transducer to a 45 degree reflector in the middle of a straight pipe, a long the pipe for maybe 8cm to another 45 degree reflector, which sends the signal out of the pipe to a second transducer. This makes a sound path shaped like a U with a long flat bottom section. This is not ideal, as there is a lot of non-laminar flow around the reflectors, but most other chamber shapes seem to have greater issues that the simple U shaped path.
Any way, why not first try to work with a sampling scope? You can buy a 11801x/CSA803x with a SD24 sampling head for under a thousand easy. That has low enough jitter to see sub 5ps shift in a pulse's edge, should be enough to diagnose the critical parts of the circuit. If you really need something real time in the end then you can just flog it again and rent a couple 10s of thousands worth of scope.
If the 100GHz scope were a few thousands dollars its definitely what you would like for this work, but obviously its not what you are going to pick in the real world. A sampler is a PITA, for a couple of reasons, although its probably the only practical tool for capturing the full resolution of the signal. When flow is turbulent you would really like to be able to capture individual measurements in great detail. You'll probably have to live with what you can get from a general purpose scope around the lab. Most of these meters run from a small battery, so they only sample a few times per second. That means a sampler will take a long time to get a detailed picture. You can implement a test mode that samples rapidly, but then you aren't looking at real operating conditions.
I'd investigate on how those flowmeters work. For sure the readout can't cost $100k+ so there must be a good way to make measurements with several $k worth of equipment.
A measly pic can measure delay with 3.5 ps resolution (50ps single shot). But unless you build it all from theory and have it work straight away in practice you'll probably need a scope at some point to diagnose where you screwed up.
The Microchip CMTU module has been around for a while, and i know some ultrasonic meter makers have played with it. I don't know any that have found it a good choice for a production meter. I would be interested to hear from anyone who has.