I do not think the dye will work due to the carpet being fully synthetic.. You have to paint it with fabric paint or just enamel spray paint may work.
I agree totally. I work for a detail shop and we do all sorts of interior dyeing. Floor mats, carpets, fabric seats, leather restoration, and so on. What we have leaned over the years is some vehicles have what we call 'Plasticrap'. The definition of Plasticrap is any type of synthetic/polymer/non-organic carpet of fabric. These are the types of fabric are very difficult to vaccum and you cant even scrub with a machine because the friction of the scrubbing action will melt the fibers and leave you one very unhappy customer. (and probably an insurance claim to boot.)
My suggestion to you would be take the best of both worlds. Remove the carpet. Clean and restore it, and Herculiner/Rhinoline the tub. I understand the money is tight as it is with everyone in the country right now, but take your time now and do it right...you will save yourself mucho headaches later down the pike.
See if these steps can help you out.
-Remove Carpet. Carefully!! Unbolt what you have to. Move what you have to. The idea here is to remove the carpet in one piece. If you have to use a looseing agent on the hardware (ie: WD40, PB Blaster, Ect.) don't worry about getting it on the carpet. We'll touch on how to remove it later.
-Vacuum and Prep tub. Remove all of the drain plugs. Set these aside. Go through and vac out the tub the best you can. Then take a stiff wire brush and remove all of the loose debris that you can. Re-vac. Now, go over the entire tub with 100 grit sand paper. Don't try to make shiny metal every where, that's overkill. Just knock the surface down so it's smooth. Re-vac again. Take isopropyl alcohol (availible from any drug store...super cheap and works better than Naptha as a surface prep agent.) and wipe the entire tub down.
-Install Herculiner or Rhinoliner as per manufactuers instructions. Allow to dry.
-Carpet time! Now I know you stated that you wanted a black carpet instead of a green one....but hey, go for a black a green theme on your Jeep. Be cost effective, and save the money for other things. Here is what my buddy and I did to my carpet, with dazzeling results. I have a tan carpet. 20 years of dirt, grime, mold and 1 quart of 10W-30 spilled in the back by the tailgate. I thought this thing was shot beyond repair. So I didn't mind experimenting on it. We pulled the whole carpet out, layed it on a clean concrete floor. We didn't even bother vacuuming it. We used a gallon of Purple Power (a strong alkali degreeser availible from most any auto parts store. Simple green would also work.) We poured 1 gallon degresser into a five gallon bucket and filled the rest up with very hot water. The key here is the hot water. We hosed the carpet down with a straight stream to remove most of the debris. Take a stiff brush (we used those brushes from Home Depot that you can put on the end of a pole. I believe that they are called deck brushes.) and the degreesing mixture and scrub the heck out of this thing. Don't be stingy with the cleaner either. Use the whole five gallons if you need to....and fair warning: There will be suds every-freakin-where!! We scrubbed this thing for about twenty minutes, let it dwell for about tem minutes, and scrubbed it for another twenty minutes. And then rinsed them. I found that if you hang the carpets in the air, we used a large A-frame ladder and some pieces of wood spanning the middle, you can rinse all of the degreeser out fairly quickly. just keep them in the same spot to dry. The fact that the carpets are 'Plasticrap' make them dry pretty quickly in the sun.
Before you attempt dyeing or painting the carpet, give this a try. You may like the results...and if you don't you just made the carpet very clean for whatever you choose to do.