It would be natural to say: “I will double-check with Mel next Monday and update you with further information.” Or, “I’ll double-check with Mel next Monday and update you with more information.” This sounds less formal and would still be appropriate.

I would say update you with further information. You could say update your information but that would not feel very natural. Update you information is not grammatically correct.

Your sentence doesn't work because it doesn't make sense grammatically. Stripped of all the extra words, your sentence is "I will update you information" ( which cannot be parsed) "your information" would make sense. To make the sentence work, you either have to either provide the verb (update) with a direct object (like "your information"), or you need to modify "update" with an adverbial clause (like "with your information"), depending upon what you are trying to say.

I noticed one more issue that no one has mentioned yet. "Double-check with Mel next Monday" sounds more natural than "double-check next Monday with Mel."

The sentence has a minor mistake, the correct preposition to use is "on" instead of "with". Therefore, the correct sentence would be: "I will double-check next Monday with Mel and update you on further information." To express the same idea in a more native way, you could say: "I'll touch base with Mel next Monday to double-check and will keep you updated with any further information."