It's not the wiring. It's the LEDs themselves.
LEDs have a characteristic delay when turning them on which may be more than you're used to from a lifetime of incandescent bulbs.
Swap the LEDs from one position to another, and the problem should move with the bulb. Replace the LED with the equivalent incandescent (temporarily) and it should go away entirely.
Unfortunately, home LEDs are still relatively new and they suffer from both this characteristic delay --- which may vary from brand to brand -- and manufacturing variation --- where bulbs from the same manufacturer may vary, even within the same batch.
The only way to avoid these issues is to closely watch online and professional reviews (from trusted sources) to pick the better-performing brands; return any product to the store that is an "outlier" in the same room -- or, wait another few years for LEDs to mature (I'm serious) and just live with it for now.
You bought LED replacement "tubes" which are called plug-n-play and allow you to not mess with the ballast -- but also require you retain the ballast. You are now married to the worst of both worlds: the EMI/RFI output of the legacy ballast, which could be considerable for a magnetic ballast, and also whatever EMI is coming from the new LED "tubes", which can also be considerable if they're cheap.
If you really want to go LED, you would be better off going with LED direct-wire replacement "tubes", and bypassing the ballast. You may want to consult the Sylvania documentation and see if they are capable of also operating in direct-wire mode; some are dual-mode like that. Don't just give it a whirl.
Otherwise pull down your fixtures, see how they are wired (instant vs rapid start, 1 vs 2 wires per lamp end) and get direct-wire LED "tubes" which are compatible with your lamp wiring. And bypass the ballast. Be sure to consult the factory to learn what kind of EMI the LED "tubes" will be emitting if any.
Plan "B" is to find a quality ballast with known very low EMI, and change that ballast, then either try your LED "tubes" or just use real fluorescent tubes. The light from them is superb these days, better than LED.
Plan "C" is that they make DC power supplies with the same formfactor as fluorescent ballasts. I scored several hundred of them on eBay for nothin'. Then get LED strips, two 5m rolls of 3528s should replace 3 fluorescent tubes, and cut them to fit in the fixture and power them off that DC supply. Obviously, the EMI output of the power supply will be the ruling factor; DC LED strips have no EMI at all unless you dim them.
Lastly, check your grounding. If your grounding is less than tip-top, it will greatly increase EMI. If the grounding is not tip-top, make it so, remembering that Code now broadly allows retrofitting of just a ground wire.
Best Answer
I don't believe motor starting flicker is the source of your problem. I have the same issue, and during my time working at a power company, there was actually one client who had this same issue after retrofitting to LED tubes.
The flicker that you notice is well below 60Hz(My assertion, not yours) and is continuous from your descriptions as the compressor runs. The discussion of the starting capacitor or a soft start is actually a totally different problem. What happens when your compressor begins its compression stroke, and proceeds to end it, is that the torque required to turn it increases steadily and then drops off sharply as the compression stroke is finished and the piston passes over top dead center.
This varying motor load results in the line current varying at a lower than 60Hz frequency (typically the compressor turns much slower than 60Hz). As the current varies up and down in phase with the compressor, the voltage drop across your wires also varies, meaning the voltage at the LED bulbs is also going down and then up every time there's a compression stroke.
For reference, my compressor is a behemoth I inherited from my father. It's rated at 5HP and draws 28 amps RMS on a 240V circuit, and still causes this problem. The fact that my pole barn is the furthest thing from the service entrance on the house (and thus the transformer) doesn't help, nor the fact that I have a 10kVA transformer. The compressor operates at about 3-4 Hz (estimated based on my hearing) and the lights flicker about that fast too.
The problem is due to poor light design (assuming your voltage flicker is in spec, and it very well may be, as my former company had done voltage studies on a customer with this problem to see if the flicker was caused due to out-of-spec flicker voltage and found nothing). I haven't torn apart any of the LED tubes I used to retrofit my house, but I suspect they forego active power factor correction circuits in order to make them cheaper, because that type of circuit should be able to compensate for this type of issue(Look up Active PFC).
Whether or not light-strip-in-fixture LED lights have better designed power supplies or not I cannot comment on because they are obnoxiously expensive($80 USD per fixture where I can retrofit a 4ft 4 light LED fixture to LEDs for 10 bucks a bulb at $40 USD!) and that has prevented me from purchasing them.