What's the best way to get rid of 1 satoshi dust?
In the current environment, I doubt that you can.  Blocks are full these days, and the block space needed to spend your UTXO, whether as its own transaction or as one input of many similar transactions, could be used instead for a transaction with a much more lucrative fee than 1 satoshi.

I think your "aggregated" strategy would only work if one of the other participants was very generous and donated a sufficiently large fee to raise the average for the whole transaction to a competitive level.  You'd be getting a free ride in that case.

In theory, one could spend the dust in a transaction having only an OP_RETURN output, either of value 0 (thus leaving the 1 satoshi as a fee) or of value 1 satoshi (thus destroying the satoshi).  However, nobody is going to mine it on the BTC mainnet, as mentioned above, and for that matter nobody is going to relay it either, since it's dust.

(Incidentally, it appears that spending the UTXO in a transaction with a single P2PKH output of value 0 does not accomplish this; such an output remains spendable (!) so it doesn't reduce the number of UTXOs.  Example on the BCC testnet.  I wasn't expecting that.)