Generally, I think the answer is "no". However, you can wait till it is really well cured, then clean with acetone and scrap the surface and immediately mate the parts with EPOXY (immediate so you don't give any oils too much time to come to the surface). It will hold together BUT you can't ever place any reliance on structural integrity. If the joint is going to be stressed then the answer is still no - don't do it. There isn't much point to creating an impermanent bond; spend time cleaning the excess glue, shaping, sanding and finishing the piece only to have it come apart on you later.
I doubt it will hold at all with PVA glue (white glue). It might hold with CA glue, but again not very well. So, Epoxy is probably your only option - IF you want to take the risk.
If it is for something cosmetic, then I would say it's probably okay. Just know that the bond will never be as good or as permanent once you have oiled the wood unfortunately. And if it for something cosmetic you might as well just use some double sided tape?
If it is for a slingshot and if you are laminating parts together with a core and brass pins, you could try peening the pins or using some sort of bolt pins. BUT again, it's probably not worth the trouble... If you are just thinking of just wood to wood lamination without any other structural augmentation, then for your own safety, don't.
You could also try a test piece and stress the bond to see how well it holds and for how long... BUT I don't think it's worth the trouble...
If it were me, I would just look for fresh wood to work with and put away those that has been soaked (label them first though). I am sure some other suitable use for it will come up in the future (says the hoarder in me...)
I think that should about cover the options in this scenario? I don't know much about cactus juice but I think it needs to seep into the pores of the wood but the pores will probably be sealed already by the oils... still, you might get some of it to take? This is just my honest opinion... Please don't place any reliance on my opinion - I am no expert!