Measuring Transect Length on Sea Floor
The easiest and larger improvement in accuracy you can implement is switching from the non-differential single frequency GPS measurement to a double frequency and/or differential GPS measurements. That usually takes down the error of single measurements from a few meters to a few tens of centimetres or less.

The GPS systems that are within cell phones, tablets and most ships calculate their position standalone and in real-time using only one of the two civilean frequencies that the GPS satellites broadcast. The greatest source of error of those measurements are the discrepancy between the atmospheric and ionospheric models and the reality.

There are two workarounds to this problem, both often used simultaneously to get even better results.

The first workaround, is two use more sophisticated GPS instruments that can record the two civilian frequencies that the GPS satellites transmit. As each frequency is refracted by the atmosphere in a different way, the GPS can better calculate its effect and correct for it.

The second workaround is to do differential measurements. That is correcting the measurements of your GPS based in the errors recorded at the same time by a nearby GPS station at a fix known position. That is call a base station.

You can install your own base station on the ground near the area you work, or use one from the international network of base stations.

My recommendation, would be that you buy a double frequency GPS. If cost is a constrain, go for an used one on e-bay and you can get an excellent quality GPS for just a few hundred dollars. Then you put that GPS to record while you do your transect and it will generate a file with raw GPS measurements. Then, later, at home you upload that file (maybe after a format conversion) to a free GPS processing service and you will get back a file with the GPS track with MUCH better accuracy than your handheld GPS.
This would be equivalent to use the first workaround I described and the processing I'm suggesting here is called PPP (Precise Point Positioning).

When you record the raw GPS data and do the processing later is called post-processing, and it is the best practice if you don't need the high accuracy measurements in real-time. After a week or so, the DoD issues high accuracy corrections of the positions of the satellites and their internal clocks. So if you wait for this high accuracy corrections you can get better precision than any real-time processing. All this is already embedded in those free online processing systems. So you just need to wait a week, upload your file and wait for the result.

The next step would be to have a second GPS and set up your own base station (or check if you work reasonably close to an existing base station). However, my guess is that the gain in accuracy won't make much sense given the errors in the relative position between your GPS and your bullrake.

A second different approach would be to equip your bullrake with some sort of distance measuring wheel, that rolls over the sea floor and records the distance traveled by the bullrake. But it might be significant technical challenges to make such a device work reliably.