Cool.
Now remember, I don't have a Mark III, only a Mark IIC, so YMMV. I'll just offer suggestions and then you'll surely see the best way to do it once you have the amp open.
Presumably, you have one of the Mark III schematics from the web. Note that these are of questionable accuracy, but should be good enough to work from to decide your plan. Unfortunately, the schematics that I've seen don't have the components labeled by name (ie "R30"), so it makes discussion difficult. Maybe someone here has a version of the schematic with the labels.
Looking at the schematic, you see that the "pull deep" is effected by a switch that, when the knob is pulled, makes a short-circuit path around a 22K resistor. When it is shorted, the 15uF capacitor hanging off the cathode of V2B is able to participate in the circuit, which causes a boost in the bass response of that gain stage (hence "pull deep"). Your new relay/JFET will do the same thing, it'll control when you choose to short that 22K resistor. You can insert your relay/JFET in one of two ways...your choice is completely based on your own opinion of how you'd like your new mod to interact with the existing pull deep switch:
Option 1) In-Series with the Pull Deep Switch: The existing pull deep switch is built into your master volume pot. It's the two lugs sticking off the back of the pot. You could snip one of those wires and insert your new relay/JFET in-series. If you do it this way, pull deep will always be defeated with your knob pushed in...regardless of what your relay/JFET is doing. Your relay/JFET will only work with the knob pulled. In other words, if either one of the control systems says "no" (either your relay/JFET or your pull knob say "no"), then you have no pull deep. That could be cool...
Option 2) In-Parallel with the 22K resistor: Here, you'd insert your relay/JFET to jump directly around the 22K resistor. In this configuration, if either of your two control systems say "yes", then you will have pull deep. It's kinda the opposite of the in-series connection above. This behavior could be cool, too.
So, it's your choice on how you want it to work. Either is totally valid. It's just a choice.
In either case, you could activate the relay/JFET by a footswitch, or you could have it completely automatic based on the R2 or LD mode. That's another choice. Footswitching gives you the most control (maybe sometimes you *would* like to have pull deep in lead mode...lots of people do). On the other hand, autoswitching is perhaps more convenient. Your choice.
So, based on this discussion, your own mind has either leaped directly to the circuit mod that you think that you'd like to do, or you need more guidance. If you need more guidance, let me know whether you think that you'd prefer the In-Series or the In-Parallel behavior. Then, let me know if you'd like it to be footswitch activated or automatically activated based on the R2/LD mode.
Again, I've never done this specific mod. But I have modded amps a fair bit (as have others on this board), so I figure we could all work together on this one to figure it out.
Chip