You'll get lot's of diverse opinions.

I'd consider cutting it to London and Paris only ..... I think 3 is a bit tight.

I'd consider going by Eurostar between London and Paris as it's civilised and city center to city center with a relaxing journey both ways. Not much longer (if it is) than flying considering transfers both sides.

You must be in London for St Patricks night. St Patricks night I'd spend at your choice of one of the many real Irish pubs ..... just google and choose eg top Irish pubs ....

A possibility is Water Poet, 9-11 Folgate St, Shoreditch, E1 6BX. Folgate street is in a lovely street where the Hugenots congregated when they arrived in London and you could see Dennis Severs House/ before your drink and meal which could be Dublin Bay prawns, soda bread and beef, oyster and Guinness pie which is often served on St Patricks night. If you enjoy Brit Art .... then you just may see a few well known British artists (world class) who live and work very close by and are regulars and possibly wouldn't miss St Patricks night ......

Enjoy your visit.