I think someone on Mathematics would be able to answer this question best; mathematicians never say bigger number. I cannot put my finger on it either but to say that, colloquially, bigger is a psychical size; larger is a quantity.
When we say "slightly higher register", we are referring to the social context in which we use the words -- not the meaning of the words. If you speak with the president of a country you will probably use "high register", for example.
-A large amount of light has penetrated through tree leaves.
-A great amount of light has penetrated through tree leaves.
Concluding that 'large' and 'great' are interchangeable but not 'big'
when it comes to indicating math or quantity.
-A big success (correct)
-A large success (wrong)
-A great success (correct)
Concluding that 'great' and 'big' could be interchangeable when
objective has no physical scale.
-It's no big deal (correct)
-It's no great deal (wrong)
-It's no large deal (wrong)
-He has suffered a great deal of stress(correct)
-He has suffered a large deal of stress(wrong)
-He has suffered a big deal of stress (wrong)
Concluding that since 'deal' is a unit without physical scale,
'great' and 'big' are interchangeable, except for the case of 'big
deal'. I think 'big deal' is a jest of incorrect grammar to highlight
the sarcastic tone.
-He is a big guy (correct)
-He is a great guy (correct but yielded a different meaning)
Concluding that 'great' can only have an equal meaning to 'big' when it comes to quantity, outside of that the meaning is 'good'
instead.
Also concluding that 'large' and 'big' are interchangeable at any
object with a physical size.
Being a native speaker I've never had to consider it in such detail,
so I think we may have to accept this provisionally. There always seem
to be counterexamples in English.
Best Answer
Of course, I can't be sure why it sounds wrong to you, but to me it sounds a bit funny because I'm expecting "it" to refer back to something mentioned previously, and the only thing in the previous sentence it seems it could refer to is "tomorrow". And asking someone to "approve tomorrow" is just silly — there will be a tomorrow whether they approve it or not.
A minimal fix would be either to drop the "it", as you suggest — which would make the sentence somewhat terse and elliptical, but which could work in a workplace e-mail — or to replace it with "this", which at least could validly refer to the entire request. But just to avoid ambiguity, I'd prefer to expand it all the way to "this request", as suggested by trpt4him.
Or, alternatively, the request could be rephrased e.g. as:
where "it" can now refer to "a day off" (viewed as a set phrase), which can be legitimately approved.
(Ps. I make no comment on whether the brevity of the request is appropriate here, since that doesn't seem to be what you're asking about and since I'm not familiar with your local workplace communications style. In some places and contexts, your example request might be exactly as long as it needs to be; in others, you might have to expand it to five times its current length to make sure it sounds polite and respectful enough.)