If you have an underground sprinkler system be sure to mark where your heads are before aerating. Those machines can tear up your heads if you run over one with a spike.
i don't know if there is a machine that does both, but one of those rental dethatcher/vertical mowers will really tear the **** out of your lawn. then you have to manually rake up the debris. they do remove the thatch really well, but take most of the grass with it.
I would think the nature of the St. Aug. runners would make it impossible.
If water penetration is the issue, I would aerate and then rake the plugs. I aerate my lawn 3 times a year (Kentucky Blue Grass in a clay soil) for water penetration. As stated above, just mark the sprinkler heads. It costs very little up here (Colorado) to have someone else do it ($25 a pop for a good sized yard), but it is also more common up here and most lawn guys own an aerator.
I remember very little about my parent's St. Aug grass in Texas, other than the long runners, but I would think dethatching would be difficult and aeration might be a better answer.
I aerated my St Augie last Fall and couldn't be happier. The timing was perfect, I ran the aerator, dropped down some Ironite and fertilizer and a soft rain came in that afternoon. My lawn is smokin' right now....all the neighborhood chicks dig me for it.