Would think the most accurate way of doing this would be to;
1. turn on the pump
2. run for say 1 minute (long enough to be sure all the water is pushed out of the tube)
2. close the pipe with a valve between the pump and the pressure sensor.
3. turn off the pump.
4. wait for pressure to stabilise (or sample pressure reading until it does).
5. record pressure reading
6. convert pressure reading to depth
7. log.
This process will eliminate the variations that would otherwise occur with the pump pushing air down the pipe and any variations in that air flow. Backpressure on the pump with it running will not be as stable as static air in the pipe compressed by the water pressure.
This scheme combines the bubbler with the static column eliminating/reducing the problem areas of both.
If there is concern about keeping the tube clear, then the pump could run all the time except when the reading is taken.
The valve could be anything simple - like a solenoid squashing the pipe, or a peristaltic pump could be used as it won't leak air when stopped. These are actually quite easy to make - a slow geared motor from almost anywhere, a disc with two to four rollers and a track with soft flexible hose - plenty of info on the web e.g.
http://en.wikipedia.org/wiki/Peristaltic_pump.
The hardest part is getting the right tubing - needs to be flexible and survive being squashed flat - silicone rubber is a frequent choice.