Rechargeable 1.2V AA Batteries vs Alkaline 1.5V?
If the battery pack just connects to the gloves, with a resistive heating element in the gloves, then the rechargeable batteries will work "well enough".

Note that 3800 mAh AA rechargeable batteries do not exist. Batteries of lower capacity with 3800 mAh labels may exist. Such batteries are often low quality.
Reputable NimH AA are available up to 2500 mAh and PERHAPS slightly more. 

An Alkaline battery is about 1.55V when new but drops rapidly to 1.3-1.4V range and then drops with use to about 1V. A NimH battery in heating use will give about 1.1V for most of the discharge period. The difference will be noticeable but probably not vast.

If you are happy to change battery packs you could use 4 x AA NimH plus an extra series resistor or (best) a current regulator. 

When the batteries are discharged below half capacity Alkalines may have high internal impedance and voltage may drop more under load than expected.

3 x Alkaline: 3V to 4.6V
3 x NimH: 3V to 3.6V
4 x NimH: 4V to 4.8V (slightly more at very start).

The fresh 4 x NimH will be somewhat hotter than the new Alkalines. MAY cause trouble but probably not.
The exhausted NimH will be the same as 3 x 1.33V Alkalines - still rather unused.
So it will be a good solution provided that the current is not too high when the battery is fully charged.