How can I determine why my lights flicker when the A/C is running?
What, if any work was done to the electrical system when you renovated and the problem started? In addition, what, if any work was done on the house in the immediate vicinity of the electrical service entrance?

Do you happen to own a non-contact thermometer (or know someone you could borrow one from?)

If this is a bad connection (and that's my guess), it will be heating itself (more or less depending on how much current loads in the house are drawing) - ie, if your microwave draws 10 amps, it's heating itself with 20 watts (10A X 2V) when you run that - if your AC draws 30 amps it will be heating with roughly 270 Watts (30A x 9V)

Lacking a non-contact themometer you can sometimes find these by touching the grounded exterior of electrical enclosures and feeling for heat. Having one means you can also look at specific parts inside the panel (but the problem may not be there - it may be in the meter box, which you generally can't open yourself anyway.)

Ideally you'd start at some point when nothing much has been using electricity in the house, or you've actually shut off the main for several hours, go around and feel/measure temperatures, then turn on the power and turn on as many loads as possible and feel/look for an area that is getting much warmer than the rest of the wiring/enclosures.

Since you are evidently comfortable in the open service panel, you can also check for voltage differences between the incoming wire and the terminal it connects to, and/or see if the voltage you measure is very different (under load) if measured from terminal to terminal or from incoming wire to incoming wire. That checks that end of the meter-to-service cables - the other end you generally need an electrician or the power company to check.

It is important to find and solve this issue before it gets worse; you're in the early stages of "how electrical fires get started" IMHO.