To view the BAT history for your Solana-connected account, navigate to your Rewards panel and find the Payout account section. 

Click the drop-down arrow and select Go to my account.

You will be redirected to the Solana Explorer. Here you can view details like token balances and transaction history.

In the Overview section, you will see the wallet address connected to your Rewards account and the Solana (SOL) balance. When an Account does not exist, it means the address has not been funded with Solana. The Solana (SOL) token is required to send your deposited BAT.

The Token Holdings section will display the Total BAT currently held on this account address.

In the Token History section you can view the transaction history for all tokens. Filter by BAT - Basic Attention to view the potential Brave Rewards deposits only. Click on the transaction signature link to view the specific transaction details.

After clicking on the transaction signature link above, you will see the Timestamp for the BAT deposit.

Scroll further down the page to view the Token Balances section. This will show you the total amount sent and deposited for this transaction. 

You can learn more about SPL-BAT here.