You can't put "that" after "in case" – it just doesn't work, therefore sentences 1-2 are wrong.
You can put "would like" but not "will" after "in case"; to express the future, we use the present. So the sentences rewritten would be:
"I take an umbrella in case it rains."
"I can book a place at the hotel in case you stay/want to stay with us."
[Not a teacher]
Student or Learner