LDO Selection criteria
What limits the maximum output current of an LDO ?

1) The LDO itself limits the current because it exceeds a certain value

and/or

2)the LDO gets too hot and limits it's output current and voltage to reduce power dissipation

Depending on the LDO you choose I expect 1) to be an issue as 2 ms is enough time for most LDOs to respond. 2) is probably not an issue depending on how often you draw this 2ms pulse. If you draw a pulse every second or less, no problem. If you do this more often then dissipation can be an issue.

A solution could be to use a large capacitor at the output of the LDO, it will then supply the current. The LDO might not like this though so check it's datasheet. But probably the the large capacitor will increase cost more than using a 1.2 A capable LDO.

The safest option is just to use an LDO which can handle the current.