Firstly, I think the datasheet for the original battery is this:
PLF 503759 D
Dimension: 58.5 mm x 37 mm x 5.0
And not the LPP 503759 DL from the link above.
LPP 503759 DL
Secondly, I have purchased and changed the original battery with this one
Volts: 3.7 V
Capacity: 1500 mAh
Dimensions: 59mm x 36mm x 5mm
Just wandering how the original and the replacement differ.
The dimensions are identical though, and the Irex DR1000S seems to like it so far.
I will report if there is anything unusual after the replacement.