What is the best train pass to get?

First, is it too late to consider coming home from Paris? Better yet fly to Edinburgh and home from Paris. You will spend a lot of time on trains back tracking. That would more sense and probably would be cheaper. You can price it but I doubt if a pass will save any money. There are discounts available for the Edinburgh and Eurostar if purchased now. I know Eurail is a travel agency that does not operate any train and, I think, the same is true for Britrail.

if your plan is Paris via Eurostar right after the conference finishes - remember the 60 to 90 minute check in time for the Eurostar. You'll see check in then security then British Border Force then French border control all before you go up to the waiting room.

It can't be rushed and people are sometimes left behind of they were late to check in.

You would need to do the maths on your proposed journeys, it is unlikely that a Britail would pay for itself on those 3 days of travel, 2 of which you can get advance fares for.
A 5 day in a month eurail may well be worth it, if you can get pass holder fares (which are, I think, 30 Euro each way) on Eurostar, as each way on Eurostar at full price on those dates is between £110 and £170 depending on train.
Then just pay single fares on the London to Gatwick leg- don't waste a day of pass costs on that journey.

One advantage of a pass would be your arrival day of 6/17 - I know Britrail will cover both Heathrow Express and the next train to York once you're at the station. Just be mindful to validate your pass at the HEX desk at the airport. In general, the pass would provide flexibility - no need to be locked into specific trains, although with peak tourism, reservations on some legs might be prudent. You need to determine how to assign a value to flexibility. I have no experience using a pass on Eurostar.

Thank you all. Great information to consider

6/29 Eurostar to Paris

As Nigel said be in the checkin line 60 to 90 minutes before departure. As St. Pancreas is very busy, I recommend being in line at the +90 minute mark. There are several things you have to clear including baggage scan and passport control.

7/2 Return to Gatwick and fly home 7/3

Fly! There are multiple nonstop flights from ORY or CDG to LGW. Tale a late afternoon flight LGW, find a hotel near it and relax before flying home.