If this move doesn't change the number of DOGE's from current reward schedule (100 bln to the beginning of 2015 and then each block 10 000 doges starting from block 600 000), then sure, why not, it's a good idea to strengthen the security of both coins by merged mining.
The only thing not to mess with is the block reward schedule, because this is the crucial factor for investing, that helps evaluate the potential price of the coin. If the supply of the coins is messed with, and the originally planned supply is changed, that would make any coin a joke (just like fiat is a joke because the future supply can be printed at whim in any quantities if needed) and investors would flee that coin in packs. Cryptos are good and different, because the coin supply increase is known for years ahead. Let's not play with that.
This would only apply if there were a real market for those coins, but there is none.
Look at what happened to DOGE since the first halving.
Did it double in price?
No, it did the opposite. Why? Because there is no market, no real demand, pure PONZI.
The only chance I see for Litecoin is to make some good distance to the competitiors by innovation.
Not by flawed assumptions whose only goal is to scam more investors out of their money.