How much data does it take to prove a given transaction exists on the blockchain?
In theory, you would only need the data from the block of the transaction, and the ability to prove it the transaction still exists after at least 6 confirmations.

The more confirmations (blocks mined beyond a transaction) the lower probability it becomes possible to reverse. Generally speaking, 6 confirmations is enough to identify a transaction is irreversible.

Here's a quote from Mastering Bitcoin by Andreas Antonopoulos:


  By convention, any block with more than six confirmations is considered irrevocable, because it would require an immense amount of computation to invalidate and recalculate six blocks. 


Block vary in size, but in SegWit transactions, the entire block can be a maximum  ~2 MB.

These days there are many online tools that can verify this without needing the entire blockchain.
The bitcoind JSON RPC API also works fairly well in providing transaction data. (Cannot be manipulated like with an online API)