"Basic question... does it usually snow around that time.. ? "
Hopefully not - but snow is possible only 12 months a year at this altitude! The lake in front of the hotel, where now is a whole tent village and a grandstand for spectators and a car park for the polo tournament and the horse races on ice, will have the lst ice plates melt away and the first fisherman trying to catch a mountain trout, as May 1st is the opening day of our fishing season.
Have you had a look at www.engadin.stmoritz.ch already? You would have a lot to do for 2 weeks also in May, so you should really be able to find something to keep you occupied for your only 1 1/2 days.
I already gave you the tip to go to Diavolezza with the complimentary montain cable cars pass you will get - this is a perfect high mountain excursion and you could extend it further to Alp Grüm (still included in your pass) or even down to Poschiavo or Tirano (paying a return ticket from Alp Grüm and back) and enjoy the whole Bernina line. I think you refer to the Bernina Express in connection with Poschiavo (not the Palm Express as you write).
St. Moritz is more a village than a "town", it's a small place and you have seen it in no time. But this is not what you are coming for actually. It is the Upper Engadin valley as a whole with it's different villages and hamlets, some of them with ancient traditional buildings in an architecture you will not find anywhere else - but what you really are coming for (at least most of the visitors are) is the nature and the surrounding mountains . This of course you can enjoy in May as well, as the mountains are not removed just because it is not peak season anymore.
A lot of Hotels, Restaurants and expensive designer boutiques and jewellers will be closed - do you need any of them? There is always something open and even more than enough for the fortunate ones visiting during this uncrowded period. If you mention to sample local food and "inexpensive" cuisine, I think you would not frequent the expensive places that are closed anyway - and then with just the 1 1/2 days, how many restaurants are you planning on visiting during this short time?
So just a suggestion:
After checking into your hotel, cross the road to the lake and walk along the lake a short stroll to the Segelclub (http://www.7500.ch/joomla/ - they may be open during May as there are many locals eating there as well). They have good good at decent prizes. Staying at the Bellaval, have dinner there on your first evening. They have a good inexpensive restaurant. and they serve local food as well. Try e.g. the local things like the Barley Creame Soup, the air dried beef carpaccio, the mixed cold meats with cheese (you can also share a plate so you can try more things). Then as you main course a Veltliner or Engadiner Rösti, a mushroom risotto or Capuns - those are all local dishes. If you did not have your fondue until then, they also have good fondues.
Next morning start early on the Bernina line and go either first off at Bernina Diavolezza and got up to the top (make sure you have your complimentary pass - it's expensive). Have a cofe or tea on the terrace and enjoy the panorama of snowcovered peaks and glaciers. Just go there if the weather is nice, otherwise you will end in a cloud. Already the ride up and down gives you a 360 degrees panorama. If weather forecast is better for the afternoon, do this in the afternoon. You find the weather forecast with local forecasts on www.engadin.stmoritz.ch.
Then continue on the Bernina line as far as you want. Stops along the line are Alp Grüm with a beautiful view to the glacier and the Poschiavo valley below with the lake. Then Poschiavo itself, which is an interesting place with an interesting history. As you mentioned in you other thread you wanted to take pictures of the train, I think you got good advise already there.
Have lunch in Poschiavo - you might already be able to sit in a garden restaurant as this is much lower and much warmer than the Engadin. Try the local specility Pizzocchery, buckwheat pasta with cheese and vegetables and butter - delicious and have a glass of Veltellina wine (the next valley, where Tirano is) with it. Also here, the mixed cold meets (air dried beef and ham) is good and it is different in each place you try it.
If you postponed Diavolezza to the afternoon, have lunch there, either on the sunny terrace, sitting on sheepskins (they provide wool blankets if it is cold) or in the panorama restaurant. They have good local specialities as well, but probably more expensive than the Bellaval and certainly more expensive than Poschiavo.
Next stop on your way back could be Morteratsch (the restaurant there will be closed however) for a hike of any lenght into the Morteratsch valley towards the Morteratsch glacier. The path will be well groomed also in May. Trains stop once an hour.
Upon your return, head along the lake and get up to the viewing platform on the lake for the panorama. Have a look at the bioculars provided and you will see how the area looks in winter (visiting in summer - respectively how it looks in summer if you visit in winter). Then take the long electric stairs through the parking and have a look at the pictures. This will have you arrive right in the center of St. Moritz and the "designer row". Stroll through the pedestrian zone, which will be rather deserted however, but there are always restaurants and bars open and people around - there are people living here as we are and we are around as well. Check the restaurants. I think the restaurant of the Hotel Hauser hotelhauser.ch/en/…breakfast-cafe.html is open as also the one of the hotel Steffani http://www.steffani.ch/de/die-restaurants , very good but more expensive.
Next morning with your half day, you could either do a nice hike to the Staz Lake www.lejdastaz.ch/hotel-restorant-staz.html the hotel will however be closed - this is a nice round trip in pure nature. Or you could have another high mountain experience, going up to Corvatsch http://www.corvatsch.ch/en/home.html . You would have to take a bus to Silvaplana-Surlej right from the train station to get to the valley station. Another option would be to travel up-valley by bus along the lake plateau to Maloja, the last village before the dramatic drop of the Maloja Pass and return for your train back to Chur.