What does your shop manual say for checking the charging system? It's relatively easy to check the output to see if there is any, but a manual will cover it in specific terms. Do you have questions with what it's saying? If you don't have a shop manual for your specific bike, buy one before you spend any more money on charging systems.
When I googled info on regulators I came up with this link:
Does your battery have a built in BMS? If it doesn't, I wouldn't use it no matter what regulator you have. Fun fact - you can kill your lithium battery by putting too many volts into it, and you can kill it by discharging it too far. Yep, discharge it too far and it's game over, no recharging. Toss it and buy a new one. A built in BMS will typically protect against that and shut things down before the battery is damaged.
Did you get a lithium specific charger with your battery, or is the charger you're using safe to use with lithium batteries? A built in BMS also allows you to use some chargers that aren't lithium specific. Even with BMS though, a charger with an anti-sulfation cycle can harm the battery.
I'm a big fan of lithium iron batteries. I COMPLETELY understand why some people are not and would agree the best bet with vintage bikes is to check/optimize everything and then install an AGM battery. (I DETEST lead acid batteries, I don't care how cheap they are). That said I won't be running a lithium in either of my single cylinder projects. There comes a point they aren't worth the cost/hassle/risk of using.
Here is the aftermath of what can happen when you put a BMS protected battery into a 30 year old Ducati with no other safeguards and then roll the dice thinking it will all work. Spoiler alert - the bike said "Built in BMS? Bitch, please. Hold my grappa and watch THIS!"
And finally - if you must jump on the lithium bandwagon make sure you are buying a lithium IRON battery, and not a lithium ION battery. It's confusing, but there is a difference. Lithium iron batteries do not tend to burst into flames like lithium ion batteries have been known to do. I'm not sure but I think most motorcycle starting batteries are lithium iron. It gets confusing though, I once bought a less expensive brand and it was labeled iron on the box but ion in the documentation!