Sorry to add to the stress of LOTS of info, but a few items above are not quite right.
"Knob and Tube" wiring is mentioned several times. This was a VERY old style which certainly did NOT include any Ground leads. I replaced such a system in our house when we bought it in 1971. But there were other wiring cable systems after that that still did NOT include Grounds. My parents' house built in the late 1940's had what was called "Loomex" wiring with no Ground leads in the in-wall cables. The wiring code in 1971 certainly required Grounds in all cables. But whatever the details, you already know from your electrician that the wall cables include no Ground wires. And YOU have remedied that for ONE outlet used for your computer.
You are sure already that the two outlets in that room are on the same circuit / breaker, doubtless rated at 15 A. Using an extension to reach an outlet in another room is NO guarantee that the third outlet will be on a DIFFERENT breaker. You would have to verify by plugging things in and turning breakers on and off. By the way, the 15 ft max extension cable length is not written in stone. The limit is mainly in the surge current at start-up. Based on the running current spec of the AC unit, that starting surge likely is in the range 10 to 12 A, and that lasts for a couple seconds. If you use an extension cord with a heavy wire gauge designed for that, you are OK. The cord should be rated for at least 15 A, which means its wire gauge should be 14 or even 12. Then you are safe with a longer cord of 25 or even 50 ft. That is, IF you can find an outlet that is NOT in the same circuit as your room.
The Surge Protector you have deals with ONE of the issues. When the AC starts up and pulls a large current for a second or so, as that current drops to the normal running current there CAN be a spike in Voltage on the lines. Similar things can happen when it shuts down. Those very brief but high voltages sometimes can damage other items on the same circuit. In your case, the PSU of the computer will be the major item at risk, but also any other peripherals with their own connections to the wall, like a printer or scanner.
The other issue is that the same surge of current draw on the lines at start-up of the AC can mean the Voltage on the lines is too LOW briefly. Most PSU's and mobo Voltage Regulator sections can handle such brief Voltage sags so well you would never know it happened. But if not, the computer (or peripherals) might go though a complete reset - they believe the power failed completely, and suddenly was turned on again. The most troublesome result of that is the possibility (uncommon, but CAN happen!) that incorrect information can be written to your hard drive resulting in one or more files so damaged they cannot be used. THIS is where a UPS - Uninterruptible Power Supply - comes in. It can maintain power to the load when the supply from the wall fails for whatever reason, or even just from a Voltage sag. The APC BX1500M unit linked above by COLGeek, while NOT cheap, is very good. It can keep your system running for several minutes if power fails completely, and certainly can deal with a sag. On the sag item, this IS the right type of UPS, called "Line Interactive". That is, the output from the battery to the load is ALWAYS connected and feeding power out. In cheaper designs it is not, and a switching mechanism must switch the load from the wall supply to the battery inverter output, which means a brief no-Voltage time. That is exactly what you may be trying to avoid - a short Voltage sag.
Do not bother with a GFCI device. Those are units designed for electrical safty, not for surviving electrical surges. Modern elecrical wiring system include Grounds to help prevent electricity from going to the wrong places, but those alone are not complete protection. A GFCI device basically constantly measures the CURRENT flowing through the Hot line and also through the Neutral line, and these MUST match within a VERY small tolerance. If not, it is presumed that some elcrtical curent is leaking out to somewhere it should NOT go, and the unit shuts off all power, just as a breaker can do. But a Breaker only cuts off current if it is too HIGH, whereas a GFCI cuts it off it it even a tiny amount seems to be escaping. It is common now in older homes that lack Ground lines for safety to install GFCI's instead on those curcuits. This DOES provide some protection against electrical currents "escaping" to dangerouus places, but it does NOT provide any real Ground. And it does nothing for Voltage surges or sags.
Before you do spend much money, though, consider these ideas.
1. Have you had any experiences of system rebooting suddenly when the AC starts or shuts down? If that never happens, then whatever Voltage fluctuations your AC is creating are being handled well by your surge protector and the PSU. That does not prove it can't. And if it DOES happen sometimes, then you are on the edge of problems.
2. You are concerned about LOAD for the entire circuit that feeds your two outlets, which is limited to whatever the breaker says - likely 15 A. It is probable that the computer system and the AC together are NOT that large, but to be sure you can measure it. There are two ways. One is to have an elecrician come in with a meter and measure at the breaker while you are using these both fully. Costs the pro's fees. The other is to buy your own Watts meter, like this type
https://www.amazon.ca/Overload-Protection-Backlight-Electricity-Electrical/dp/B09BQNYMMM/ref=sr_1_2?dib=eyJ2IjoiMSJ9.7kti57Zt-qCEmR9lr5AFUPp_IAECpyUUZDCqggW0b7lh966kcjA3TEFO1TkswL9n3wI8oOJn0HeYpUP7SX-MGT5-m8C-HEgKox4xvrXGQDE2oyeacIm4F03l0Pi7T3sMXKxCbX3M5Kw2sYxP1XpvfSet-8IrWPmMkQRKwcO3RON9d5Sa72X7zqtfis39YriExmYxlrV3cFb6tuRYGhyy0yF6IP_Po2Trach3kT5bt_4YMH2cGtJviSVGQRhgVgwVQHsRzur753SisDCId9eAEZyQX-MCZtNouJBbkTPJeEw.C3Kp9wXS2Ux5TkE8IyheAAt1ts_MbYUMSB1F63bQEog&dib_tag=se&keywords=power+meter+outlet&qid=1714275479&sr=8-2&th=1
Not very expensive. You plug that into one outlet, then plug into the meteer the load(s) like all your computer system fed through the surge protector. Then do the same for the other outlet with the AC system. THEN find out if any OTHER outlets nearby also are on that SAME curcuit, and measure their loads, too. ADD up the WATTS for everything. A common circuit at 15 A and 120 VAC can supply 1800 Watts max, BUT in general one wants to load a circuit continuously not more than 80% of max, or 1450 W. Now you KNOW the load, and may find you worry less.
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