You should go, not too far from you surely. As for refreshments (and always best to carry energy stuff) that's what panniers are for. We are lucky to be cyclists. It's amazing, encouraged a neighbour to go with kids the other week, they really enjoyed it. Quieter than the quietest possible sunday as no tourists. Short of a nuclear attack, will never be like this again. Police won't bother you, royal parks are being sensible.
Use gps auto routing for a dawdle back and you will doubtless discover by accident lots of interesting corners in a fascinating city.