Determine a suitable replacement battery cell for an obsolete product?
NB The following comments and 'recommendations' relate to a battery used with a prosthetic device. While the advice is believed good it is offered with the usual internet device disclaimers [ACNR, YMMV, IANAL, do not spindle fold bend staple punch mutilate, ...]. "Burning man" experiences are greatly to be avoided :-(.

Does "being phased out" translate to - "not being available AND no alternative being sold by manufacturer"? Is so, multiple customer action may improve chances of a better response. However, if a higher capacity battery is being offered then it MAY be a better solution.

Any cells of the LiPo / Lithium ion type that have adequate capacity would be suitable.
The originals that you cite are well specified in that data sheet. Their capacity is small by modern standards, (1320 mAh) and typical charge current of 660 mA = C/2 is modest. 

The Digikey battery that you give as a possible replacement datasheet here is very slightly larger dimensionally so may or may not fit. It has substantially larger claimed capacity (2000 mAh vs 1320 mAh), and much longer claimed cycle life.
The stated normal charge rate of 0.2C/400 mA is below the original's C/2/660 mAh but well within the allowed C/1 / 1A charge rate.

Importantly when changing LiPo battery capacities in equipment - the new cell's recommended charge terminate specification is C/100 (0.01C) in standard or fast charge modes. Lower terinate currents = "harder" charge, and C/100 is a very aggressive charge specification (max energy shoe-horned into cell) and not conducive to maximum cell life, and it is likely that the existing charger is not as aggressive in its charge termination.
This means that, as long as the cells fit mechanically, then the new cells are probably suitable replacements because: 


Existing charger end point current is probably 'gentler" than allowed.   
While existing charge rate is probably higher than nominal recommended it is below the max allowed.
The new cells are of usefully higher claimed capacity and longer claimed cycle life. 


You do not say whether the original charger has cell balance capabilities. If not, the ability to occasionally check cell balance and  to remedy imbalances would help lifetime. 

If you can INCREASE charge termination current to C/10 or C/4 (and C/2 is better still) you will get a longer cell life at minimal loss of capacity - and still well above early capacity. 

Again - all care, no responsibility. I'm happy to comment further on or offlist due to the important application.