メニューもう少し見てていいですかって英語でなんて言うの？

You are politely requesting that you retain the menu. That is enough. You do not need to explain that you may wish to refer to it later during the meal. That is obvious.
Could you....? = This is a polite way to make a request.
Is it OK...?= This is a polite way to ask if it's alright to do something.

1. "leave the menu with me'- means that you would like it for longer or for it not to be removed.
2. 'hang on to'- means that you would like to keep it for a some time.

oh, sorry could I keep this for a bit longer?
Meaning= can I keep the menu for a bit longer please

I can keep the menu please?
Meaning= can I keep the menu on my table please

Can I keep this please? I might want another drink!
Meaning= I want to keep the menu on the table because I might want to order another drink/dessert

These are easy phrases to use in restaurants.
Although yes it is normal to take menus away after ordering,
waiters and waitresses definitely won’t mind leaving it with you.

I worked in a British restaurant for over 5 years
so it really wasn’t a big deal to ask for the menu to stay on the table. They will be happy to help

Can you leave one menu here? - you can ask the waiter to leave one of the menus on the table so that you can keep looking at the different orders

e.g. We'd like to order 2 tako yaki's, 2 pork gyoza, and the sashimi box, but can you leave one of the menu's here? We might want to order more food later.

this means that you still want to use the menu and you can call the waiter once you are done.

In the first statement, you have started with the reason why the waiter should collect the menu later. You are still using the menu.
.So, you may say:
I am still using the menu. Please collect it later.

The informal phrase 'done with' actually means 'to be finished dealing with something so that you don't have anything to do with it anymore'. In this case you are still dealing with the menu.
So, you may say:
I am not done with the menu yet. Please come for it later.

In the last sentence, you are politely requesting the waiter to let you use the menu for a bit longer straightforwardly.
So, you may say:
May I use the menu for a bit longer please?

As a customer your desires and requirements are of paramount importance
to the restaurant you are visiting. So if you want anything, simply ask!
There are NO RULES regarding menu access time;-)
While some can make "snap decisions" , when it comes to food; many of us like to "dally a while" and "examine all the options available" to us.

If you were at a restaruant and you are aware that they will usually be taken away after ordering, but you would like to keep the menu with you to look at it some more, you could ask the waiter/waitress any of the following to ensure this: "Please can I keep my menu a little bit longer so I look at it some more?", "Can I keep the menu for a little while longer?" or "Could you leave the menu with me please?".

Most waiters or waitresses will give you more time to look through a menu if you ask politely. For example you could say "May I have some more time to look at the menu?" or "Could I please look at the menu for a few more minutes?" You could also just state the fact that you are in need of more time, like "I need a little bit more time to look through all the options."

Can I keep the menu to have a look, please?
I would like to keep the menu to have a look at it later.
Could I keep them menu to have a a look at it later, please?

The waiter or waitress you say this to will certainly understand what you want if you say one of these sentences or questions. Good luck with using them.

I hope this helps.
Have a great day.
Will

1.Please leave us with one menu.
"Leave us with" means that you would like one menu to
be left behind so that it can be used by everyone at the
table in case there is something additional that they'd like
to order.

2.We'd like to keep one menu.
This means exactly the same thing - that if one menu is
kept behind, then everyone at the table can share the menu
to decide on dessert or additional beverages (drinks).