Without the specs of your induction heater it's hard to give you any answer. But if it works with your battery pack through your existing BMS and if the specs are correct, we can assume the battery is providing less than 10A max at its nominal 12V voltage. The "Maximum working current: 5A-8A" is not very clear, maybe the BMS supports only 10A peaks and 8A continuous output?
Based on these figures, if you want to bypass the battery pack, you would need an AC-DC converter able to supply 10A continuous. Many Chinese ones are completely over-rated and even if they can output their max rating (really rarely the case) they will get pretty hot. To be on the safe side you could use a 15A or 20A power supply.
If you want good quality, get a Mean Well switching PSU, like those meant for LED driving:
https://www.meanwell.com/ Most have somewhat adjustable output voltage, so a 12V one could be possibly cranked to 12.6V (but more on that below) If it's just for pass-through power then leave it at 12V. But they are probably way above your budget...
So alternatively for around 20€ you could get one of those Chinese PSUs, just select 12V and 20A:
https://www.aliexpress.com/item/1005007045758794.html
With a PSU you should in theory be able to remove the battery completely and hook it directly to your heater and get proper pass-through power.
Now for charging your battery faster, I don't have enough info to safely advice you. It's not clear what's inside your BMS and if it has any real charging IC or if it's merely passing the input current to the cells and just monitoring for an overvoltage condition. The second specs mention a balance current but it's very low so I assume that MAYBE it's the balance discharge current? And then MAYBE that BMS can do balance charging.
If the BMS acts as a proper charger then it will limit the input current to a preset value anyways, so it doesn't matter if your PSU can provide more. It will perform CC-CV charging at the rate it's setup to operate (most can adjust the CC part by modifying a component value, but since it's likely surface mount I assume it's beyond your skills level)
If the BMS is not a real charger, and your schematic kinda makes me think it could be the case, then indeed you should stick to their 12.6V 1A PSU recommendation as written on that schematic and not take any risks.
Anyways all the disclaimers are there for a reason, the dangers are real. If you over-charge your battery it could burn down your house and I don't want to be responsible for that. And even if you remove the battery from the equation, the listed 20A Chinese PSU is hazardous with it's exposed terminals, touching the live input and you'll get AC zapped.