How can we check the number of local sats in the channel?

Before it's closed you can see the balance on each channel in Thundehub or RTL. If you mean after the closing happened, I don't think there is a log of that available in Thunderhub or RTL. Your client is checking as soon it detects the channel closing transaction. If the other party were to close with a state that is not the latest your client will issue a punishing transaction and you get the entire balance to your on-chain wallet.

For RTL, Click "Peers/Channels", "Closed", then your local balance for each closed channel should be listed under "Settled Balance".

For LND run:

$ lncli closedchannels

"settled_balance" shows your local balance when the channel closed.

You can find this in channel.go where the settled balance is defined as the local balance at the time of channel closure.

func NewUnilateralCloseSummary(chanState *channeldb.OpenChannel, signer input.Signer,
 | 	commitSpend *chainntnfs.SpendDetail,
 | 	remoteCommit channeldb.ChannelCommitment,
 | 	commitPoint *btcec.PublicKey) (*UnilateralCloseSummary, error) {
    ...
    closeSummary := channeldb.ChannelCloseSummary{
        ...
        SettledBalance:          btcutil.Amount(localBalance)
        ...
    }
    ...
}

Thank you both. I was searching for a log of the balance at the time of closing after the channel had been closed. It sounds like a moot issue if there is an automatic punishing transaction if the final state is not used.