Charge Controller for Small Lithium Ion (Lithium Titanate, LTO) Cell
You are looking for chargers and/or BMS adapted for "Lithium Titanate" aka LTO cells, though I've never seen one that small before! 

They are even less common (so far) than Lithium Iron Phosphate.

What's not to like : in bulk, only the lower capacity per kg (which isn't a real issue here, but makes them less attractive for cars). Currently also more expensive per amp hour capacity, but over 20000 cycles they pay you back handsomely for that.

By reputation they are also a lot safer (less fire risk) and work over a wide temperature range. IMO they could be a game changer where weight is less important and lifetime is, e.g. for power grid storage. I discussed the economics a bit here and that has only improved since. If you get 2 cycles a day out of them (for the morning and evening peak demand) the economics of grid storage start to look attractive.

Here's a slightly bigger one, with quite a lot of information. These cells seem to be popular in China, perhaps for grid applications. BMS adapted for these are available.

I have only limited experience so far, but : 
Treat roughly as Li-Ion except for the voltages. That is : 


charge at constant current (CC) until the terminal voltage, then CV until current has fallen to some fraction of the CC value, then cut off charging.
Discharge until undervoltage cutoff point
Don't float charge or leave discharged
Ideally, balance cells with a BMS (maybe not so important with such small cells)


Limit voltages : 2.8V charging. Most sources say 1.6V discharging, this says 1.8.

Charge cutoff point : Russell is effectively suggesting 100% of the CC value which will be safe, simple, and probably fine for your aplication, but 50% or 25% should give you a few more % capacity if that's worth the complexity.

In practice there's not a lot to be gained above 2.6 or 2.7V, or below 1.8V and using these should reduce wear and only lose a couple percent capacity. Most of the action happens between 2.3 and 2.0V, so treat "2.4V" as a little optimistic, and 6 cells is a better match for a 12V (car) battery than 5 cells.

And to answer your actual question : if you augment your suggested charger with current detection (measure voltage across the current limit resistor) and some means to cutoff charging when current falls to (your choice of value) you should be fine. (I would add a timeout: say 4 or 5 minutes since 20C is 3 minutes, as a safety measure if practical).