With a 1st class Eurail pass you can travel in Standard Premium, not in Business Premium, on the Eurostar. You can also travel in Standard with a 1st class pass.

You must have a global Eurail pass to travel on the Eurostar and there are no additional fees apart from the cost for the seat reservation fee. Please read more about Eurostar reservations below. 

My advice is that you check availability on the Eurostar before you buy your Eurail pass.

There is a limited number of pass holder seats on the Eurostar and on popular departures and during high season those sell out weeks, and sometimes months in advance.

The best place to see the availability of passholder seats and make reservations on the Eurostar is 

https://www.b-europe.com/EN/Booking/Pass#TravelWish

Press "later trains" and eventually you will find the next available connection. 

If you have a mobile pass you need to generate a Pass Cover Number in order to make the reservation at b-europe. You do that here in the PCN generator:

https://community.eurail.com/news-and-announcements-39/pass-cover-number-generator-is-live-5653

As far as I am aware there is no upgrade option for pass holders on Eurostar so you would need a full price ticket for Business Premier compared with the 38 euro first class pass reservation fee for Standard Premier. I believe the only real benefits are a better meal (It couldn’t be worse than the last one we had in Std Premier) and the use of a lounge before departure. 

The main benefit of Business Premier is the quick check-in and the fully flexible ticket allowing change of train at any time with space on any other train guaranteed. No doubt mainly used by people on generous expense accounts.

IN the UK with a 1st cl pass you have access to all lounges when they are in stations anyway-but in that St. Pancras station it will hence be the local one for Midland trains-there is also one in KingsCross just across the road. And then you still have to adhere to the MIN check-in times etc. Most of that time will be spent standing in long Qs for all the checks and controls anyway, though I dont know how bad that is also for 1st. compared to 2nd

I can assure you that Standard Premier travellers have to check in along with standard travellers. The difference is. as on many trains, more space for passenger and luggage and level of service onboard. Business Premier does have priority check-in but so few travel in Business Premier it has no impact on others. At least they have a guaranteed quiet lounge once checked in until train arrives.

I haven’t used Paris E* but Brussels post check-in gets very crowded and can be difficult to find seats nearer departure. Lille is a lot smaller and they don’t start check-in until about an hour before departure, so the queue builds up quickly - expect up to 30 mins standing to get through to waiting area unless you really are at the head of the queue. Lille Europe is a dismal place before check-in and E* waiting area not much better.

Note that once checked in you cannot go back out into the general public as you have passed passport control so station lounges are of little use,to departing passengers. They are though very useful for onward travel after arrival (First Class LNER lounge in Kings Cross has been well used as we wait for our train north after E* - they even have a short route to platforms that can allow you to board train before others are allowed)