Can you safely triple the brine recipe for corned beef?
You should be fine scaling this recipe up, as long as you are sure to scale all components equally.

The USDA regulations for commercial brining and curing give a maximum of 200ppm (parts per million) sodium nitrite in the finished product.  They also stipulate a minimum of 120ppm ingoing nitrite for adequate preserving properties in refrigerated products.

Let's use the metric weights for your formula to calculate its concentration by itself:

Weight of single batch: 4000g(water)+350g+225g+42g = 4617g

Weight of sodium nitrite: 42g*6.25% = 2.625g

Concentration of brine = 2.625/4617 = 568.6ppm

Now, that may sound like a lot, until you realize that commercial brining solutions often use concentrations of around 2000ppm.  For example, federal standards also say that 2 lbs. of sodium nitrite can be used per 100 gallons of water, effectively a solution of about 2400ppm, assuming it is injected into the meat at a rate of 10% per original meat weight.  (That's only nitrite and water; once salt is added to the mix, the concentration would come down to around 2000ppm or somewhat less.)

How is this in compliance with the USDA?  Because very little of the solution is generally absorbed by the meat.  Even when directly injected into corned beef, usually the meat only gains about 10% by weight, which means those concentrations effectively drop to 1/10th in the actual finished product.

For a relatively short cure of a large piece of meat (as in your case): if you wanted to be in compliance with USDA preservation standards, you'd need to weigh the meat before brining, weigh after, calculate the weight gain, and then calculate how much solution was absorbed to determine whether the nitrite falls into the 120-200ppm range.  My guess is that your brine wouldn't even hit the low range of 120ppm in the meat unless you injected it.  (With longer curing or smaller pieces, more nitrite could circulate in the meat, so we'd have to do a different sort of calculation then, which would assume that the solution was coming closer to equilibrium with the meat; but that won't happen in 7 days.)  Nevertheless, even lower concentrations of nitrites will add significant preservation qualities, even if they don't hit commercial levels.

In terms of toxicity, you also need to factor in the chemical reactions which happen in the meat (and produce that pink color).  Some nitrite will be converted into nitric acid and bind to other components of the meat, effectively rendering it harmless.  So even if you calculate the amount of nitrite solution that was absorbed by the meat, it may not given an accurate representation of how much is actually left in the meat once various chemical processes occur.  (And by the way, the lowest published toxic dose for humans is 14mg/kg and the lowest fatal dose is probably somewhere around 25mg/kg.  You'd probably need to eat your entire 15 lb. brisket cured at the maximum commercial brining level in one sitting to get near that amount.)

Finally, in terms of efficiency, I doubt you should need 3 gallons of brine for a single 15 lb. brisket.  The Culinary Institute of America's Garde Manger book has a recipe for corned beef involving 3 gallons of water (and 198 grams of pink salt, for what it's worth, higher than your concentration).  But it's for 4 briskets of 10-12 lb. each, and they even do an injection of 10% of the meat's weight before submerging.

Thus, I doubt you'd actually need to increase the recipe that much in an appropriately sized container (maybe 1.5-2 gallons at most?).  Also, if the brisket is oddly shaped and doesn't fit well, I might consider cutting it into 2 or 3 pieces that will fit better and require less brine.  That will also increase surface area and absorption, probably approximating the results of the recipe you found for the 5 lb. brisket.