2-3mm lateral shift means either a loose cup (DS cup in pic appears properly seated) or some serioisly worn out ball bearings. If it’s bearings there will be more damage to the spindle.
As Spago said,unusual for a cup to back out later in life.
Literally ANY italian threaded 113-115 mm Shimano or Shimano compatible BB will interface just fine with that crankset. UN55 is better and there are more italian threaded ones out there (checked eBay and they are out there for like $30 plus shipping)
Velo Orange sells one that I think is rebranded Tange and they are also good.
Spend more or less, your choice. Personally, I would buy the needed tools and do it myself if my LBS was unable or backed up.
__________________
Hard at work in the Secret Underground Laboratory...