No, that is unrelated to Tier Point earning, and is somewhat misleadingly written. There was a time when Avios earning had a floor for most airlines so you would earn at least 300 to 500 Avios on short flights, bearing in mind mileage was the underlying root of the Avios system. Now that Avios earning is mainly cash based then that minimum rarely applies if you are buying inexpensive tickets. As I say Tier Point earning is an entirely separate process, and so that 300 Avios statement is a red herring as far as your plans are concerned.

If you buy a BA codeshare you are on the BA system anyway, not Aer Lingus, and all revenue flights get at least 5 TPs. If buying on prime Aer Lingus just double check the fare bucket to make sure it is one mentioned in the website as being an earning bucket - most of them would be.