There are lots of hotels in the area. Look on Booking.com for something that suits your taste and budget.

With such an early train the next day I’d look for a hotel near Gare de Lyon. No idea of your budget but Ibis Gare de Lyon is very comfortable and convenient. Paris is a very safe city in the tourist areas. I would take a taxi from the taxi rank to get to the hotel. Only use licensed vehicles and they always have a meter. Otherwise use the RER to cross Paris but you have to change . As you are likely to be tired, probably don’t speak French I don’t think this is a good idea.

thanks for your advice. Is there any chance to get the reservation for seats at the train station which has shown not available seats via Eurail website?

That depends on what train you want to take. Eurail only sell the 10 EUR seat reservation for the TGV trains. They don't sell the 20 EUR seats. Until the TGV is full you can get the 20 EUR seats on several places.

What route do you want to make a reservation for?

Around both big stations are dozens, dozens of HTLs-also in many price-levels. Around Nord is IMHO much worse area as around Lyon, so with the early dep. I would stay around G de Lyon (or Bercy=just side by side)-booking.com is usually the site that shows most with most ways to choose-but there are several otehrs. IBIS is a  chain from FRance with many HTls there and always a safe bet. Ibis BUDGET chain is however really for penny-pinchers.

from Charing Cross London to Intelaken OST Switzerland on 10/08/2022, we’re holding Eurail pass in1st class for adults & teenagers.