I took a similar trip two Aprils ago. Loved the trip. I flew into London and stayed at the hub hotel at Westminister St James Park for a few days, then took the Eurostar train through the Chunnel (warning, some people on this forum don't like the use of that term) to Paris. I have a favorite hotel in the 6th in Paris, that is practically perfect in every way - Hotel de l'Abbaye. I returned via train (through the Chunnel) to London and stayed in the hub at Covent Garden for the weekend to see West End shows. It was a fabulous trip. The hub hotels would probably not work for you because they are small and even the bigger room would mean a shared bed. They are so inexpensive in price though, the two of you could each have a room and still be in your moderate budget. Or you could check out the regular Premier hotels. I loved the locations of both and the 6th is my favorite in Paris. It would probably make more sense for you to do an open jaw flight, flying into London and out of Paris, but I had already made my reservations using points. I had perhaps by favorite afternoon tea at the Conrad around the corner. A lovely trip!

hub hotels -
https://www.premierinn.com/gb/en/hub.html

Paris hotel, but less moderate in price -
https://hotelabbayeparis.com/en/over...hoCV-MQAvD_BwE

Afternoon tea at the Conrad's Emmeline -
https://www.viamichelin.com/web/Rest...unge-15ard9ids

A wonderful idea. It would help to know a concrete definition of how much money you would classify as moderate.. Almost every hotel you might be interested in will be near a tube or metro stop. What's your style?

Thank you so much. I will make note of those hotels and we plan to fly into London and out of Paris. My husband is traveling with us so there are actually three of us.

$200-300 per night. We have stayed in B and B's in the past. Just like clean, pleasant staff and good location.

Oh, 3 people definitely impacts your hotel room choices.
What are your interests in each city? Historic sites? West End shows in London? It's hard to go wrong in either city if you choose a central location.

Oh, lovely! Well, you probably want to go to the Portobello Market on Saturday then.
Portobello Market | Opening Times & Information
I would send a guide book to your granddaughter to let her decide what she really wants to see. Also decide which West End show y'all are interested in. Hint = Hamilton tickets are a LOT less expensive there if that's of interest. If so, listen to the soundtrack first to be familiar with the lyrics of the faster songs. On this last trip, my "sites" were Kensington Palace and the Royal Mews - which my mom and I almost saw on our first visit but ran out of steam. It was my first stop on this April visit. I loved it and your husband may as well - and of course, you and your granddaughter. I would decide on one "big" site a day and then fill in around those. I filled up my Oyster card and used the tube on the first two days but it was cold and rainy so I took black cabs the rest of the time. My preference anyway. I like to SEE the city instead of riding underneath it. So, 3 big sites in London in 3 days + nice afternoon tea, a train trip to Paris, dinner at Le Bonaparte and two days exploring the main things she is interested in Paris. That would be a nice, easy paced trip. I'll share some photos of the Portebello market day. It was fun, even though it was cold and rainy. I stopped at the Scotch Shop on the way in to buy a warm shawl.

300 USD = 271 euros
300 USD = 232 pounds
Certainly doable

The Hotel de Siene in the 6th has a triple for some dates in May for 250 euros (refundable).
Great location.

I'd strongly recommend reading the lyrics (in addition to listening to the soundtrack) before you see Hamilton. It's a fantastic show and much better if you understand it! Lin-Manual Miranda himself published the lyrics.....Hamilton-Lyrics-ACT-I1-e82b4f261a.pdf

I agree.
When the show was new, there were videos with lyrics on YouTube but they were taken down. But I just checked and the show is back up with lyrics.

I guess the gave LMM proper credit (or sought prior approval) this time.

I needed the lyrics for Guns and Ships.

This hotel in Paris was perfect for 3. We had a nice bedroom and our granddaughter had her own space with a pullout in the sitting room. Very nice indeed. Read the reviews...
https://www.tripadvisor.com/Hotel_Re...de_France.html

In London we stayed here twice. Both times with a granddaughter. Perfect for 3. Bedroom with a bathroom in between that and a sitting room with a pullout. We liked it so much the first time, we went back with a second granddaughter a few years later..2014 and 2018.
https://www.tripadvisor.com/Hotel_Re...n_England.html

taking the Chunnel from London to Paris is very fast and easy.

I would take TPAYT's advice to the bank.
In fact, I would book both hotels right now and change later if you find another option you prefer. I'm going to bookmark for a future possible trip with niece and great-niece.

Also agree about the easy train trip through the Chunnel. A great experience!

starrs.....thanks for the vote of confidence but many others, with more experience than I, can give more choices.
I will say that these hotels, style and location, worked perfectly for us and our granddaughters.
Park Plaza Westminster Bridge..London
Residence Hotel Henri IV..Paris

Also, some of the best food (breakfast & dinner) we had in London was at Park Plaza. Breakfast every day and dinner, 1 night each time, when we tired from days activities and didn’t want to go out.