I have consulted an electric engineer, who took a look at the MR-1 battery,
so first some more info on this. The battery cell has the following identification (img 1):
Sony Fukushima SEW
US473850 A8T
02B80F30G
We searched for this cell on the web, and it seems that the cell is used in many different applications. However, it looks like it is only possible to order the cells in packs of at least several hundred pieces. So, I think they are not meant for end users.
Another important point the engineer mentioned is that when replacing the two cells by new ones, it is very important that the replacement cells are a "matched pair" (you probably know this principle from matched Mics for stereo recording).
The other part of the MR-1 battery is the small electronics, and the codes on the chips are (img 2):
G2PN
WB 6E2
We couldn't find out much on the web on these - maybe someone else is more lucky.
So now to my battery replacement. A while ago I had purchased the battery
Replace NP-120
for Fuji FinePix 603 3,7V, 1800mAh/6,7Wh, Li-Ion
(Replace DB-43 D-Li7 NP-120 BP-1500S)
as a replacement battery in the MR-1.
The egineer encouraged me to try to replace the two battery cells and put instead the single cell form the battery NP-120 - on my own risk, of course.
So we took the battery apart, and he mounted the new single cell on where the old two cells where mounted (img 3). As the old cells where mounted directly in parallel onto the electronics, mounting a single cell instead should be possible. After the egineer had soldered the new cell onto the electronics of the old MR-1 battery, we packed the new cell back into its original case, and put that into the frame of the old cell (img 4, had to cut the original case in length a little). Then I mounted the battery into the MR-1 (img 5) and tried it out. It worked.
A test recording with the fully charged new internal battery gave me a recording time of 1h25Min. With the old battery I only had 3 Minutes! So up to now the replacement seems successful.
I played in a concert last week and took a recording - worked, and another rehearsal session during the week - worked. Up to now I'm quite happy with the replacement.
But I should also say that I never would have tried this without an experienced person.