Li-ion battery charging termination current
It would be helpful to have a data sheet of this battery, however bad it is.

0.2C charging appears excessively conservative. This would mean over 5 hours for a full charge (most likely 6-7 hours). I’m not familiar with every possible Lithium-Ion chemistry, but I hadn’t ever heard of a chemistry requiring only 0.2 C charging; note that it might be beneficial in minimizing battery wear, especially if such a long charging time is of no concern to you. Thus, it would also be helpful if you pointed out what you’re optimizing for: battery wear, charging time, running time to discharge the battery, etc.

Given that 0.2C is 0.82 A, then your battery is supposedly a 4.1 Ah battery. Typical guidelines indicate charge cutoff at 0.03C to 0.05C. Looks like the 0.03C figure was chosen to approximately reach the 0.12 A recommended by the manufacturer.

Assuming this is a run-of-the-mill chemistry like LiCoO2, and your charger cuts off at 10% of the programmed charge current, I would think you could go for a bit over 0.2C charge current, so that the cutoff falls in this 0.03C to 0.05C. Given that 0.5C is generally safe for chemistries I’m familiar with, I would personally go with somewhere from 0.3C to 0.5C so that charge is cut off at 0.03C to 0.05C. Thus, 1.2 A to 2 A would probably represent a good choice; choose from the lower end if you want to be conservative and minimize battery wear, and from the higher end if you can’t afford slow charging times.

Again, note that I’m assuming this is a run-of-the-mill chemistry and may not necessarily apply to your battery’s specific chemistry. Hence, this is why a datasheet would be most helpful to give a better answer.

Regarding a lower current cutoff “more fully charging a battery”: yes, you may get a couple % extra by stopping the charge later. However, this reflects negatively on battery wear. Batteries wear more the longer they stay on a high charge level. If it is easy and cheap to replace the battery, it may be a valid tradeoff. On the other hand, on my laptops, tablets, cell phones, I personally avoid charging them to 100% to try to extract some extra lifetime from their batteries.