Ok, I'll row in with my 2 cents worth. If it were me.....
First thing to do is step back and think.
It was running fine then you drained the battery.
You swapped out the battery for a new one. There is no issue with it being a lithium - it's just a battery. It either has enough power or not. I'm running a lithium in mine at the moment.
We know the new battery is OK as it turns over the starter motor (normally - as per your first post). This is the highest drain on the battery, so I assume it's good. The fact that the instrument cluster lights up is another indication.
So, the only change is the draining of the battery and swapping out to a new one. This leads me to believe the possible sources of the issue are
1. The draining of the battery caused the ECU to log a fault code, which is still present and causing the bike not to fire. And this code isn't being reset with the presence of the new battery, but must be cleared by a GS911 or motoscan. I find this unusual as it would mean that any battery drain or swap would leave you stranded on the side of the road. I also didn't need to reset anything when I swapped out my battery, nor did I get any fault codes. but I swapped out a good battery for a good one, so not exactly the same conditions. So hold on this one for a while.
2. When you swapped out the battery, you either disturbed or didn't reconnect a lead. I know (from putting in the hot swap mod) that there is a live from the battery to the starter relay (which is the relay pictured above by Moto Vasya 21.) This relay is fine as the motor is turning over as per post 1. I don't know if there is another lead from the battery to power the rest of the bike, but that's the area I'd check first. After all, that is the only area disturbed between the bike working and not working.
3. Outlier possibilitiy. Leaving the bike on "park", i.e. with the key set to steering lock and lights on, caused something to burn out/overload that is not affecting starting. Unlikely but a possible point.
4. Second outlier, there is no gas in the tank Unlikely, as you said there is, but I'd doublecheck. If it's cranking over and you're not getting a spark, and there is gas in the tank, and the injectors are spraying, then you should get a pretty strong smell of fuel from the exhaust.
5. One that just struck me, You didn't by any chance switch the Kill switch off. Do check as that would be a laugh (and you wouldn't be the first). If I remember correctly, this would allow the motor to turn over but not fire.
5.a After reading a few posts, it looks like the kill switch is prone to failure on the K1300's especially if it has been left out in the Sun. Try turning it on and off a few times with the ignition on, to see if the fuel pump can be heard cycling.
So, before I'd go down the rabbit-hole of taking things apart and/or swapping components, I'd take a good long hard look at what was disturbed, i.e. the battery area, and check that everything is as it should be. Then I'd check with a gs911 to see any error codes.
Then, and only then, would I start taking things apart as you could cause more problems.
Other points - a service indicator, i.e. "your bike is due a service in <date>" will not stop the bike from working.
Do come back and let us know how you get on.