The issue is not formal speech/writing versus casual speech/writing; the issue is how given or given that is functioning grammatically in the sentence.
Given is a preposition -- a word that typically combines with a phrase to form a modifying phrase. (Examples of common prepositions = between, among, under, and on; when inserted into a prepositional phrase, they modify a noun, as in the book is on the table, the book is under the table, the book is between the sofa cushions, etc.)
The preposition given means "in view of" or "considering," and it appears at the start of a modifying phrase (not at the start of a complete sentence): Given the high cost of gas, I had to stop driving. I canceled the reservation, given his allergy to shellfish.
Given that is a compound conjunction. Conjunctions join parallel elements -- in the following examples, the conjunction joins two complete sentences: He bought a laptop so that he could work from home. She feels much happier now that she has moved out of her sister's flat. The boss deemed the purchase unnecessary given that she planned to sell the firm.
Parts of speech are not interchangeable. You can't use a preposition (a modifier) where a conjunction (a joiner) is needed. Both of the examples you shared require given that (conjunction) because both are composed of two complete sentences that need to be joined: Given THAT I’m tired, I won’t be there. I don’t like you, given THAT you’ve been mean lately.
I've been a copy editor and a writer for 30+ years, and I checked reference books to make sure that my post is accurate (it is).