Can you just “upgrade” and rebuild the engine instead of having to do the recall?

You'll probably have to provide a lot of documentation/proof that the valve springs have been replaced. If you're planning on rebuilding the engine, just get the recall done first. Stupid, but looks like the only option at this time if you want to register your vehicle.

Just found out today I couldnt renew my registration on my '13 due to the same issue in California.

My registration is due on 10/28/20. Does this mean I have 1 month to get my vehicle repaired and registered?

I'm not sure what to do at this point as I was not planning to get it repaired

Do what just about everyone else did in CA. Get the recall done or you can't register it. You've had over a year to get it done. Numerous threads like this one have been stating that the recall is mandatory in CA for a while now.

Yes after 10/28 you won’t be able to legally drive the car. You may be able to get temporary registration but they may not give that to you with the TSB still being open on your car.

Also, there are fees that you have to pay if you don’t register the car as inoperable.

Honestly, it doesn't take long to get done. Find one of the better dealers to do it and you should have your car back in 1-2 days. Serramonte kept hounding me to get it done and finally when I went to renew my regs I got the notice that I needed to provide proof of getting it complete.

When your recall is done make sure to get the salmon/orange certificate as proof (or get one from your friendly dealer), send it in with your payment and then you are all set to wait for your tags to come in.

DMV has been slow af processing things. I didnt get my tags for 2 and a half months after I sent everything in. Sooner the better.