First of all you need to calm down - 1h05min is more than sufficient to transfer to a Z gate- even from ASW bus arrival (no problem). You will make that transfer unless your incoming flight is seriously late.

As I recall it there are hardly any lines for passport control when you walk to the Z gates. You don't need to clear security check again. No need to book any transfer service as you seem to know your way around FRA airport.

For your information LH819 has arrived on time or even early for the last 7 days (only 1 day it arrived 12 min late, but that is not going to ruin anything)

There are a lot of options to go get DPS if you need to get rebooked. TG via BKK, QR via DOH, EK via DXB etc. LH will select what works best for you.

Most likely you will make that transfer so no need to take into account the risk of misconnecting at FRA. As I as said before; you need to be realistic and calm down. You are almost certain to make this connection even without running/being in hurry.