Converting cordless to corded: Pulsing issue
You are confusing amp-hours and amps. A 1.3 Ah cell can supply a LOT more than 1.3A - try 13A or even more if the screw is stuck - albeit for a lot less than a whole hour! In this case, theoretically 6 minutes (0.1 hour) but practically, probably a bit less, depending on the condition of the battery, the temperature and so on.

So the problem isn't that the PSU is switch mode - the problem is that it senses overcurrent, and shuts down, and then tries again to see if the overload has gone ... rinse and repeat.


Find a good (or even marginal) battery pack. (Or dismantle two and replace dead cells with good cells from another). 
Rig the battery pack to run the tool externally via a cable, like you're using the PSU.
Measure the actual current taken
Use a PSU rated at that actual current.
It may be possible to rig a lower current PSU as a trickle charger, continually replenishing a dodgy battery pack which provides the bursts of current you need.