Andrew, it is possible to run all control code from the rpi, but a lot of work will need to be done.
I have just started working on modifying the apm code for more functionality and autonomy, with good results so far.
See Recomended Complete ROV Control package? - #4 by jwalser
If you are interested in the servo hat, search for pca9685 on ebay, it can be had for less than half the price of a servo hat. The apm is worth the price just to have all of the necessary hardware and sensors on one board.
I have considered both routes deeply, and I think going with the apm is the better option at this point.