What you're up against is the infernally small size of US electrical service - 1800 watts. European countries get between 3100 and 3600 watts at any random receptacle. However, Americans have an ace up our sleeve that'll let us bring up to 4800W to a receptacle.
Ganging both circuits to serve one receptacle is out of the question. It is a bad practice for a dozen reasons, unworkable due to GFCI, and utterly impracticable absent an isolation transformer rigged in a way that would really need engineering supervision (i.e. Your institution has a staff of electricians directly overseeing this setup, as such, I won't go into details.)
There are several ways to solve this.
Turn off the other loads on the circuit
Seriously. Are you 100% sure you know every outlet on the circuit and are ypu positive there aren't any other loads also burdening the circuit? Because if there are, it would explain everything.
Bigger cables the whole way
You have the right idea with a 10 AWG extension cord. You just need to go all the way back to the service panel with it, so you are not relying on that long run of skimpy 14 AWG built into the house.
More likely than not, there is a receptacle directly underneath your service panel. This is the electrician's outlet, when he wired the house, he "hot-wired" this receptacle so it was energized when nothing else was. And then he ran extension cords for his tools and lights. When he was nearly done, he punched it down into its own circuit breaker. Good chance it's 20A. That is a great receptacle to use.
Rewire the faraway receptacle for 240V; get a transformer.
America does have a standard for 240V and it's as good as Europe's. It's just rarely used. Doubling the haul voltage cuts current flow in half, and cuts haulage losses by 3/4. 15A at 240V transforms to 30A at 120V - more than enough. In fact 16 AWG extension cords would suffice if operating at 240V. The goal would be move the transformer as close to the work as practical, though your excellent 10 AWG extension cord will be perfectly adequate to haul 120V long distances, don't alter it.
Assuming the long-run receptacle you use is 15A, I'm not sure whether it's MWBC or not, it makes no difference. Cap the neutral, convert that circuit to 240V and stick NEMA 6-15 outlets there.
If it's dual 14/2 or 14/2/2, even better - those are two separate circuits, leave one 120V and make the other 240V by taping its white wire on both ends and connecting it to a 2-pole 240V breaker of same amperage. And fit a dual NEMA 5 and 6 receptacle - they make those. Likewise you would change the ends on an extension cord to NEMA 6.
Now for the 240 to 120 transformer. This is a little tricky. The cheapie step-up/down transformers are dodgy because they don't isolate, and the "common" side is presumed to be neutral. In North American 240V, both legs are hot, no neutral. On the other hand, these things come with a Schuko plug, which is non-polarized, so in Europe you never know which leg will be neutral.
A better choice is a true isolating transformer, such as a 2 KVA 240/480-120/240 transformer meant for small service installations. Jumper one side for 240 and the other for 120, obviously. Since this secondary is isolated, a single ground fault will not kill you. A GFCI on the output would be nice, though.
The ideal choice, hard to source in the USA, would be the 110V style of power used in UK construction sites. They use an isolation transformer, as I described in the last paragraph, but the 110V side has a center-tap, which they ground. As a result, even if the tool has a ground fault, it is only 55V from earth, which is not likely to be lethal. I don't think you could simply import a UK construction transformer, becuase they are made for 50-cycle operations.
