Electrical – Lights start flickering on Luminous inverter when I connect the Desktop CPU

This is almost certainly from using a "modified square wave" - MSW - inverter rather than a "true sine wave inverter" - the power conversion circuits in AC LED fixtures expect a sine wave input, and the way they behave when fed MSW input is upsetting your inverter.

One possible solution, short of "buy a new inverter" (a rather expensive proposition, especially for true sine) would be to feed the light circuits (only) though a small isolation transformer. The inverter power would feed into one side, and you'd feed the lights a much better approximation of a "true sine wave" from the other side. It should be a bit oversized for the load, since the MSW input will cause some heating of the transformer that a normal sine wave would not. However, the load is so small that almost any isolation transformer will be a bit oversized for it. Find the minimum size by looking at the VA (not W - power factor matters, a lot, here) rating of your lamps, and then run a fudge factor for MSW input of about twice that rating.

Unless the VA is absurdly bad for a 1.2W device, something like this for roughly $50 ($40.76 plus shipping) will probably work. You'd also need to mount it in a safe enclosure, provide circuit breakers or fuses, etc...

My APC UPS has a setting to be more tolerant of the input power. I found it necessary to enable this when running from a generator during power outages. Without the setting, the behavior was just as you describe -- constant switching between line and UPS power. Read the documentation for your UPS carefully -- there is probably a similar option.

Edit:
Here's an excerpt from the APC BE450G UPS manual:

Voltage Sensitivity Adjustment (optional) The Back-UPS detects and reacts to line voltage distortions by transferring to battery backup power to protect connected equipment. In situations where either the Back-UPS or the connected equipment is too sensitive for the input voltage level it is necessary to adjust the transfer voltage.
1. Connect the Back-UPS to a wall outlet. The Back-UPS will be in Standby mode, no indicators will be illuminated.
2. Press and hold the ON/OFF button for 10 seconds. The OnLine LED will illuminate alternately green-amber-red, to indicate that the Back-UPS is in Program mode.
3. The Power On/Replace Battery LED will flash either green, amber, or red to indicate the current sensitivity level. Refer to the table for an explanation of the transfer voltage sensitivity levels.
4. To select LOW sensitivity, press and hold the ON/OFF button until the LED flashes green.
5. To select MEDIUM sensitivity, press and hold the ON/OFF button until the LED flashes red.
6. To select HIGH sensitivity, press and hold the ON/OFF button until the LED flashes amber.
7. To exit Program mode wait five seconds and all LED indicators will extinguish. Program mode is no longer active.

The details for other models probably will differ.