sounds like the battery and voltage regulator are both working properly...
but my short answer is to verify that the starter is a 6v starter and not a 12v starter. which was the root cause of all my misery... (hint) 12v starters typically have four brushes, 6v starters typically have two. if you're luckier than me, information on the casing will answer that question. the loooong story follows for the those that dare.
i had pretty much the same problem. engine would turn over the engine (not quickly) and start the car when cold, but when warm would barely turn it over, sometimes not at all. the battery would be dead after only about 20 seconds of cranking, max, before needing to be put on the charger.
i replaced all the battery and ground cables with 00 copper wires and cleaned up all the connections too, but still didn't seem to fix the problem. there was an old starter in the trunk of the car that bench tested successfully so i removed the supposedly new starter that was recently put in the car by the previous owner, and THAT fixed the problem, although i'm not convinced that there was anything physically wrong with the new one...read on.
i can't be sure, but i think that when the previous owner began messing around with an 8v battery (which was in the car when i purchased it) and tweaked the voltage regulator too, they not only burnt up a couple of gauges on the dash, ruined the voltage regulator, but also installed a new 12v starter (there are no markings but there are four brushes which is usually an indication that it is a 12v motor) that didn't, doesn't, won't and never will work properly with either an 8v or 6v battery. why did they install a 12v starter, i'm not sure and will probably never know, but it could have been in error or it's possible that they just didn't know what they were doing.
in any event, my car now has all new 00 copper battery and ground cables, a new 6v battery, a new 6v voltage regulator, some new gauge (all of them now work), and wiring under the dash and its original 6v starter and works wonderfully to start the car, whether cold or hot. and can crank for quite a long time (NOT that it needs it, mind you!) before the battery loses power.
it's good to keep in mind when dealing with electrical circuits, that heat increases resistance. meaning problems are more likely to show up as temperature increases.