It is definitely very formal if you use the noun 'restroom' instead of 'toilet'. Using the noun 'toilet' is also formal but gives the obvious impression that someone is very much being pressurized by nature to go and relieve himself/herself of waste matter. If you use the phrase 'need to', it gives the impression that the person certainly has the urge to go to the restroom. But, if you use the phrasal verb 'feel like', you mean to ask the person if he/she seems likely to go to the restroom.
Instead of using the noun 'restroom', you may also use 'ladies room' for women's toilet and 'gents for men's toilet.
So, you may say:
Do you need to go to the restroom?
or
Do you feel like going to the restroom?
or
Do you need to go to the ladies room / gents?
'toilet'と言う名詞を使う代わりに'restroom'を使ったら、これは間違いなくフォーマルです。
この 'toilet'と言う名詞を使うと、これもまたフォーマルですが、誰かが本当にトイレに行くことに迫られていて早く用を足したいという印象を与えます。
この'need to'（する必要がある）と言う動詞句は誰かが急いでトイレに行って用を足したい、と言う印象を与えます。
ですが、もしこの 'feel like'（～したい気がする）と言う動詞句を使うと、トイレに行きたいような気分がする、という様な意味になります。
この'restroom'（お手洗い）と言う名詞を使う代わりに 'ladies room'（女性用トイレ）や 'gents'（男性用トイレ）という事も出来ます。
ですので、次のように表現してみましょう。
Do you need to go to the restroom?
（トイレに行かなくて大丈夫ですか？）
Do you feel like going to the restroom?
（トイレに行かなくても大丈夫ですか？）
Do you need to go to the ladies room / gents?
（女性用トイレ/男性用トイレに行く必要がありますか？）
Would you like to visit the restroom before we leave?
Do you need to use the water closet (WC)?
Here are three ways you could ask the question!
"Do you need to use the restroom?" (This is the most common)
"Do you need to use the bathroom?" (This is informal, used more with kids.)
"Do you need to use the water closet (WC)?" (This is used commonly in Europe.)
Another thing to ask is....
"Would you like the visit the restroom before we leave?"
You would ask this if you are leaving the place where a restroom is located. Such as a restaurant, business, etc. You would also ask this before leaving on a trip.
次の3つの質問の仕方があります!
"Do you need to use the restroom?" (最も一般的な表現)
"Do you need to use the bathroom?" (カジュアルで、よく子どもに使われます)
"Do you need to use the water closet (WC)?" (ヨーロッパでよく使われます)
別の表現には以下があります。
"Would you like the visit the restroom before we leave?"（行く前にお手洗いに行っておきますか？）
レストランやお店などトイレのある場所を離れる場合にこの質問をするでしょう。旅に行く前にもこの質問をすることがあります。
１）Would you like…で確認すると、礼儀正しく丁寧な印象を与えられます。お手洗いは、私自身が米系企業が長いため、bathroomまたはwashroomを使うことが多いです。
２）ただ、ビジネスの場などで「お手洗い」とハッキリ言いたくない場合、「休憩しない？」という言い方でやんわり伝えることもあります。
You might want to go to the restroom because there is none where we are going
"Do you need to go to the restroom before we go?" this is a formal way of asking someone if they need to go to the restroom.
"You might want to go to the restroom because there is none where we are going" this is a formal way of telling someone to go the restroom but you are giving them a reason as to why you are telling them that.
"Do you need to go to the restroom before we go?"
行く前にトイレに行く必要はありますか？
これは、相手にトイレに行く必要があるか尋ねるフォーマルな表現です。
"You might want to go to the restroom because there is none where we are going"
行くところにはトイレがないので、行っておいた方がいいですよ。
これは、相手にトイレに行くように伝え、どうしてトイレに行く必要があるのか理由も伝えるフォーマルな表現です。
Due to the lack of toilet facilities at the venue, may I suggest we take advantage of those available now - if we need to.
If there is 'a lack' of something' it means that there is a deficiency of something.
"Due to the lack of young, fit adults of working age, many elderly people now find themselves forced to carry out strenuous labour."
「lack」は何かが不足していることを表します。
例：
Due to the lack of young, fit adults of working age, many elderly people now find themselves forced to carry out strenuous labour.
〔訳〕若い元気のいい若者が不足しているせいで、多くの年寄りが骨の折れる労働をせざる得なくなっています。
When you want to ask someone if they would like to use the toilet; you can say:
-Do you need to go to the loo?
-Do you need to go to the restroom?
-Would you like to use the loo?
トイレに行きたいかどうかを尋ねたい時は、こう言うことが出来ます。
例文
-Do you need to go to the loo?
トイレに行く必要がありますか？
-Do you need to go to the restroom?
トイレに行く必要がありますか？
-Would you like to use the loo?
トイレに行きたいですか？
There won't be any bathroom facilities where we are going. Would you like to use the restroom now?
Because you want to be formal the word "restroom" is the better than the word toilet. You can use one of the follow to ask your question.
1) Are you in need of the restroom?
2) There won't be any bathroom facilities where we are going. Would you like to use the restroom now?
"Would you like... ?" Asking a question with "would you like to?" is a polite/formal way to ask "do you want to?"
"restroom"の方がフォーマルな言い方なので toiletよりもいいです。以下の例を使うことができます。
1) Are you in need of the restroom?
トイレに行く必要がありますか？
2) There won't be any bathroom facilities where we are going. Would you like to use the restroom now?
今から行くところにはトイレがありません。今トイレに行っておきたいですか？
"Would you like... ?"は"do you want to?"（～したいですか？）のフォーマルで丁寧な表現です。
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