Unusual bottom bracket shell
I'll put some thoughts that appeared in the comments:


  I've googled and came across someone else with similar problem: bikeradar.com/forums/viewtopic.php?t=12592913 - there is a cross-section of this bottom bracket:
  
  Looks to me like a French invention.
  
  I wouldn't try to remove the "fixed" cup. Just like you said - clean everything, find a new axle and assemble it back again. If you remove the whole set you end up with a threadless BB frame and those are said to be not worth to be brought back to life.


Suggestion of Batman:


  Depending on what the threadless BB frame is like, you can use threadless BB's like the velo orange grand cru one


which I agree to, however the price tag on Velo Orange bottom brackets is hefty. It may be worth if the frame has substantial value either financially or emotionally to you.

From my own experience - I have a bicycle with high emotional value which has Thompson BB (pressed in cups). The BB shell is 42mm in diameter. Velo Orange has a solution for that however it exceeded my budget. I was lucky enough to find a press-in Thompson/Thun replacement of sealed bearings cartridge with a square-taper spindle (the original was cottered, hence the idea of replacing it).

In the meantime I came with an idea for you. Since it is a road bike, perhaps it would be possible to upgrade the BB to, let's say, Hollowtech? You'd have to measure the inside diameter of the BB shell (dark blue in the cross-section, labelled as Frame) and check if press-in Hollowtech bearing cups exist, maybe some adapter shells will be necessary.

Another idea is, if you want to stay with some vintage yet standard stuff is to press in a complete BSC threaded shell:

This one I've found here: http://www.wiggle.nl/sram-pressfit-30-naar-bsa-adapter/ - seems quite "generic" and available in more countries.

Then, when having BSC threads you can simply put sealed bearings cartridge with square-taper spindle and enjoy the maintenance-free bottom bracket.

But again - my choice would be to find a new spindle (since you already have a square taper) if the spindle is of standard measurement, replace the bearings and assemble it back again - there is only one cup and cone to adjust and, as I can see in your pictures, there are the lip seals on the BB spindle making the whole package quite sealed, thus maintenance-free.