This is a bit jumbled, but I hope it helps.
First the good news....
There is also the original Griffin (OS), the Open Source/Reprap version. Both it and the Pro were designed to be scalable.
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griffinprinter.org]
The only other delta truly designed to scale well that I know of is the Wolfstock, which is better for sizes as large or larger than the Griffin XL. The XL though, is pretty much the largest Delta (and cartesian for that matter) you can buy as a kit (or assembled) for anything less than a new or used car and about the biggest for anything even remotely close to your budget. Deltas (and cartesians) can only scale so much before your parts are extremely bloated or extremely fragile.
Now some bad news...
Considering your plans, I would forget the Mini Kossel right from the start, not because it's a bad printer or that it doesn't scale well, but considering the amount of weight you will be moving around on the effector, a Mini Kossel will probably be too flexible, especially if you try and scale it. While the Ultibots version uses 20x20, it still uses a lot of the Kossel upper frame which isn't meant to take that much moving mass and dragging a feeder tube around. An issue you may face with the Rostock Max is the amount of wood involved, it uses A LOT, which could catch fire. That isn't to say a Griffin won't either, like the Mini Kossel, it still uses cast (Pro)/printed corners (OS) that are not metal. While the Makergeeks All Metal solves the fire problem, being based on the Rostock and relying on linear rails for the frame, it will not be a very stout machine (I had an original Rostock). Realistically, just like the Kossel Mini, the Griffin XL is probably too flexible for what you want to do with it but the Mini and Std would be good. The mini is one of the most stout printers around.
This is just the tip of the iceberg in terms of issues you will face.,
Due to the heat coming off, and flying debris, don't count on belts lasting long, in fact belts may not be a good idea at all (they too could burn). There is also a question of torque and mass, considering the amount of weight you want to move, standard Traxxas rod ends are probably not going to work, besides wear and tear they could melt.
Now the dream killer...
It's not even going to be remotely close to being within your budget. Large format is NOT cheap to do right, it can be done cheap, just not right, and this alone pretty much destroys your budget. If you just wanted to make a small welding printer above, you could probably do it for a few thousand. Throw size into the mix, you just doubled your cost. Add in the fact that you haven't built a printer before, double it again. While Nema 23's aren't badly priced, the stepper drivers needed to drive them properly are, they alone will suck up a large portion of the budget you set.
Using a delta may also be a bad idea... While it can offset the need for larger motors, it can only do it for so long and as you go larger, keep in mind that deltas become less and less space efficient while cartesians become more space efficient. Then there is calibrating... A cartesian is linear, you lift this end x amount, it goes up that amount. Deltas work in a bowl shape, the further out you go horizontally, the more accurate your calibration needs to be. For example, most consider a bed accuracy of around .01mm across the build surface good, in order to get that on a Kossel Mini sized delta, software calibration will probably reach it using a diagonal measurement in the .001mm range. To get that same bed accuracy Griffin XL sized delta, your diagonals need to be calibrated own to the .0001 range, if not more. It can take 10x longer to calibrate an XL than a mini, if not more. Trying to get a 4foot build plate accurate down to even .1mm may not even be within the firmware's ability.
I'm not saying what you want to do is impossible, however, you need to be realistic. Large format is still relatively experimental (and expensive), metal printing with a welder is highly experimental (and needs lots of custom metal parts I.E. expensive), and you have zero experience even with a normal printer/knowing what will or won't work. And you want to do it with a delta, which complicates things even more... I'm not saying your experience won't help, but it won't keep you from making mistakes and large format simply costs much more.
My advice, find a printer you like, buy or build it, enjoy it, learn it, then start planning how to build a larger printer or metal printer, and then finally combine the two. While this sounds time consuming and expensive, it will actually save you money because you will have a more knowledge. It's cheaper to make mistakes on a smaller printer and what you save on the large format will easily pay for the smaller printer. Not to mention you will get a printer you can use now to help build/prototype your next ones, and believe it or not, you will end up at your goal sooner rather than later.
All that said, don't expect it to be a couple weeks or or even a few months, what you want is cutting edge, and you have to learn the basics first. Before you even head down this path (metal printer), you should understand that what those guys were doing, while technically metal 3d printing and impressive, it's not what most would consider very useful yet. It's more about experimentation and pushing limits than it is about getting something out of it that you can actually use. You could use it with the expectation of putting it in a milling machine after I guess but that's something you should keep in the back of your head, after all this time, effort and money, you may not get anything physical out of it.