I'll add some here that hopefully can help you think through this. I just went through a somewhat similar situation, just I went about how to do this differently. We have a 16 year old (15 model year old) truck with 120K miles on it. It does every pulling need and weight carrying need we have, just it has a mid life of miles on it, some body and frame rust that needs to be dealt with and it is well, close to 16 years old. The cost to replace this truck with new is a lot of funds. I know in time that will come, but I wanted the ability to get another 5 dependable years out of it (another 100K miles out it) before new time comes.
The need is, cross country travel with a reliable tow vehicle that will not leave us in dire repair needs at the mercy of full shop costs days away from home. This distance is a lot more than yours, but the need is similar. Here is my post on the forum. Maintaining an older camper tow vehicle (pic heavy)
In my case, I have the ability to do my own work and a shop to do it in. To hire this level of work out at shop prices is not practical. I do not have a complete total yet in parts cost, but it is close to $5K by the time I will be done with it. Part of this is dealing with rust from living in the midwest with road salt winters. Your truck if it lived in AZ all it's life, may not have that issue. I feel good about my restoration project that I now have a reliable cross country tow vehicle for a lot less cost then new and or even a 3 to 5 year old one.
In your case, if you trust your mechanic and he says $2K will make the truck tow worthy, that is not a lot of work for parts and labor compared to anything you can buy used. I would not really trust getting a 3/4 ton tow vehicle with the engine size you have for anything less than $2K or even double or triple $2K, maybe even 5 times that. Getting a good sound ready to tow 3/4 ton truck with enough engine for your size camper is going to be in the $10 to $12K starting price and it may be 15 years old. $15K may buy you a well kept, well maintained vehicle from a private party on a year 2000 to 2003 ish 2500HD Chevy with 4 wheel drive and a 6.0 liter engine. Newer and the cost keeps going up and even then, is it tow ready or does it need all new fluid changes and other 100K mile replacements?
A question to the mechanic, if you spend the $2K, what truck life can you expect going forward with say, 3,000 towing miles per year? 1 year, 3 years, 5 years? Also a need is, what does the detail list call out? What I'm getting at, spending $2K on an older truck to get a reliable tow vehicle for 3,000 miles a year towing is not a lot in the big picture. But it also all depends on what that quote includes. Front end work can cost a good amount in parts. Then there is brakes, fluid changes throughout the truck that come into the ability to be good for towing etc.
I know the $2K may be a lot of money right now, ask the mechanic, does all the work need to be done at once or can part of it be now and then part of it 1,000 to 1,500 miles from now? What would the cost split be? This may help spread the cost but at the end of the year or early next, then you may have a reliable tow vehicle.
Mileage, in the big picture towing mileage is not going to change a lot. Maybe 2 to the most 4 mpg on a 5 year old 3/4 ton gas truck verses yours. It is just going to take fuel to tow a camper of your size. The newer truck may ride better, have more get up and go, but the fuel is not going to change a lot. Mine only gets 7 to 8.5 MPG towing, a brand new one is not going to get a lot more in large gasser engine. Maybe they hit 10mpg but, that is not a constant given large mountains and wind pulling my size camper.
Hope this helps