To directly answer your question: no you cannot "boost" a circuit.
Every wire has a voltage drop, and according to NEC it can't be more than 5% at the outlet. Voltage drop is proportional to the length of the wire, resistance (which is dependent on the size of the wire), and current. In other words, as your load goes up, so does your voltage drop.
If your existing wiring is just on the edge of either load or size, then the current may be going up enough that it's below the UPS threshold and thus the UPS is switching to back-up power because it sees it as a brown-out.
If the cable is too small (typical is 14AWG for a 15A circuit, 12AWG for 20A) then the only thing you can do is either increase the cable size, or run a new circuit (and really, you're not going to pull out the old cable and run bigger cable, it's just as much work to run a new circuit so you might as well do that).
If the load is too high (eg, you're close to 15A) then the only thing you can do is run a new circuit, or decrease the load.
You could start by using a Kill-a-watt or similar device to measure power consumed by your PC, printer, TV, etc. This will help guide you to what is consuming too much power, and where (if you go this route) it makes sense to split the circuit up.
You mentioned lights; using LED or CFL bulbs may be a stop-gap to save some power, and is certainly one of the easier things you can do.
If it comes to it, there's ways to be creative about how to split the circuit up, without necessarily having to rip out a lot of drywall. You may be able to run a new cable halfway, and then splice in and split the circuit into two, starting from a light or receptacle. You may be able to find the wire underneath in the basement, and cut it and put in a junction box so the circuit from that point on is on the new circuit. It's hard to be specific here without actually being in your house.
I have noticed that GFCIs seem to be worn out by heavy, prolonged loads. I can't find any authoritative sources for this effect. Probably the GFCI manufacturers know all about it.
The bridge circuit which detects the flow imbalance depends on some precision electronics which are heat sensitive. The heavy current itself does not affect them, but the side effect of that current heating up the components will lead to premature failure.
Best Answer
There are a variety of pre-made products which do just that. The controller sits inside your electrical service panel (or on a single circuit) and you place a "clamp meter" around the hot wire in question. It then communicates with you via powerline signaling or WiFi (cute little WiFi antenna pokes out of a knockout on your service panel.
Obligatory warning that nothing about being an electronics guy prepares you for the requirements of mains electrical (and in fact mis-trains you). One cannot just hack random electronic components into a mains electrical system, even if the components are UL listed. You may only use approved, preassembled equipment that has been UL* listed as equipment for use with mains power, and you must install it according to its labeling and instructions**. These items are not found at Mouser electronics, Amazon Marketplace, and damn well not Alibaba. They are found at big-box retailers (though they can be sloppy) and your local electrical supply house. You also cannot mix class 1 and low voltage wiring.
* or similar testing lab. CE is not a testing lab.
** because the listing is contingent on installation in that manner.