Please do not activate the pass in advance. That's not necessary to make reservations and it prevents mistakes because if you change plans and you forget to change your start date, or if you wake up ill, you will lose travel days. You can deactivate the pass on My Pass.

Reservations in the UK are free of charge and optional (except for night trains). You can book on the GWR website or at a ticket office in the UK.

Reservations for London - Paris are €30 in 2nd and €38 in 1st class. You can book on www.raileurope.com.

Also see this page about where and how to book reservations:

https://www.seat61.com/interrail-and-eurail-reservations.htm

If that doesn't work for you, then please mention travel date and departure time of each train you want to book.

La fecha del viaje es el día sábado 27/01 desde LIVERPOOL LIME STREET (United Kingdom) a PARIS NORD (France). a las 11:43 am.

Cada reserva en total tiene un costo de 43 USD por persona.

Me gustaría saber si existe una opción mas económica.

There is no train alternative to Eurostar, no. It has the monopoly on international trains from London for now (through the Channel Tunnel). 

The only alternative would be a flight, Flixbus or a ferry -> all significantly more expensive and less convenient than Eurostar.

This route is very popular, reservations cost 30€ and there is a passholder quota (pretty annoying I agree but we can't do anything). Book ASAP the 16:31 train out of London St. Pancras (use Rail Europe to avoid fees). You're lucky that there are seats left ! Passholder seats can sell out days or weeks in advance sometimes.

Reservations are mandatory on Eurostar (+ other high-speed trains in some countries), you'll get denied boarding without one.

You can use any train out of Liverpool before, simply log it on Rail Planner before boarding and hop on. Reservations optional and free.

Have some margin in London as it's a 10-15 min walk between stations + you must arrive 45 min in advance for security and passport controls (unique in Europe).

Questions welcome. (Where are you heading after Paris ? It might be worth booking the TGV high-speed train a few days in advance depending on where you're going.)