Since I don't know what is inside the supplied PSU can't answer, I know the mobility scooter charger was a stage charger, so maybe the original is what is required. The Lidi has 4 charging rates at 12 volt, 3.8A, 3A, 0.8A and 0.1A once it hits the 0.1A it shows battery as fully charged.
The whole idea is to charge slowly, if you charge too fast the chemical change can't keep up so water is lost, and if really fast it can buckle the plates, so if your in no hurry the slow charger is better, but the stage charger comes in many flavours.
Pulsed charging is also used, this monitors the voltage decay to adjust charge rate. The Lidi charger is called 7 stage, some of the stages I would not call stages.
1) Monitors battery (checks if leads on right way around) I would not call that a stage.
2) Delivers pulses until battery voltage hits level one.
3) 3.8 amp until voltage hits level two.
4) 3 amp until voltage hits level three.
5) 0.8 amp until voltage hits level four.
6) 0.1 amp until voltage drops to a lower level.
7) 0.8 amp until voltage hits level four.
To my mind it is 5 stage at best and since the first pulse stage is normally not activated I would call it a 4 stage charger.
The Lidi uses fixed amps and the voltage varies, most are the other way around, the voltage is steady and when the amps drop to a pre-set level the voltage drops, for a battery you can top up between 60 and 300 Ah it will normally start flat out, say 35 amp, as the volts hit 14.8 it will hold that voltage until the amps drop to around 4 amp, then it drops to float charge voltage of around 13.2 to 13.4 volt.
The problem arises where the battery is used and charged at the same time, if for example you have a boat or caravan on shore/site power if your lights are one you could use more than 4 amp so the charger will not drop the voltage as battery becomes 85% charged, often there is a timer as well, but the battery can get over charged, so solar panel and alternator inverter chargers use the pulse method. With a narrow boat with two 70A alternators feeding an inverter you can be charging the batteries at 140A, OK often 300 Ah of batteries, but at that output over charging can do a lot of damage.