I don't think you need PID or PWM, just a thermometer and a relay should do fine I would think.

If it takes one minute to heat the tub one degree, and it cools at the same rate, then you may keep the temperature of the water between... say, 77 and 78 degrees (any two arbitrary points, really) by turning the element on for one minute every other minute. I suppose you might want it to take two minutes to heat the water or you might want to keep the water within a half a degree of optimum, but it will be rare to find a thermostat accurate and consistent enough that a 10 second or so cycle won't keep the water very close to optimum temperature - and this only if it takes only 1 minute to heat or cool the tank a degree. If it requires 10 minutes, or 30, then turning on and off the element in 10 or 30 second cycles should keep very close temperature control.

My guess is that you will need far more watts to get anywhere near a rate of increase that requires PID or PWM and, if I'm wrong and the water does heat too quickly, you could use a lower wattage element.

I don't often take baths, but even in that small tank I am comfortable for at least tens of minutes with no auxiliary heating, but then... perhaps lavender scented bubbles retain heat?!