The existing battery is an LTC-7P that was hard to get hold of, there are still some stocks circulating but they are either expensive or not able to deliver outside the US.
So I had to look for a replacement, there is an equivalent from EVE with the same form-factor the EF651625 but tbh I didn't know the brand and couldn't find any reviews on the internet so I opted for a SAFT LS14250CNA. It has slightly more capacity but got a cylindrical shape, this particular one fits and th pinout kind of workout since the board only needs 2 connection rather than the 4 on the previous battery.
#2: remove battery, as you can see the negative pins are connected together so a 2 pin axial battery would work perfectly as long as the pitch is smaller than 35 mm and the diameter is around 15mm. if it gets bigger than that there is risk of touching the casing.
The battery here seems to be holding (3.7V) but as the datasheet shows the curve is pretty steep at the end, so difficult to know for sure its state. These are supposed to have a shelf life of 10 years, with a 34 yrs old oscilloscope and an unknown history this is pretty much a disaster waiting to happen.