Do you have a multimeter?
A simple test light might be ok too

 

I do have a multimeter

 

So it still starts straight up first go when the starter relay is bypassed?
If so, we move to the coil side of the relay

Can you check for voltage at the coil?
Most likely the coil side is 86-85 on your year too and check socket for pin 86 for power - but don't have a wiring diagram here.

Next up: the ASD relay.

Edit: coil as in coilside of the relay

 

Now that I have tried it a few more times, jumping the starter shows no distinct improvement over just turning the ignitition normally. Auto shut down relay sounds like it might have potential how do you suggest I look into that?

 

Yes
It sounds like you're having an issue 'upstream' fro the starter relay, causing you to loose spark or fuel.

what brand fuel pump did you put in? And is there a possibility for you to verify fuel pressure?

What you can do in the meantime is replace the ASD relay with a known working relay and check.

Next thing would be to bypass the ASD relay in a similar way you done with the the starter relay.

 

How is this coming along ? - have you looked into this further?

 

It sounds just like what my 96 XJ 4.0 does every now and then, like once every month or two. As long as I keep my foot on the gas, it will stay running. Once it warms up it’s fine. No codes.

My first thought is an EGR sticking open, but I haven’t dug into it. Whenever it acts up, I have to get to work so I just run with it.