Weird crank no start, then crank and start issue
My 2002 Kj is having a little bit of an issue which started about a month ago. Wife's car - no mods just basically a stock auto 02 Liberty Sport with the 3.7L V6. A month ago she tells me it has trouble cranking (misnomer sadly) so I dash off and replace the off-brand battery with a good one. It tested weak, but not dead. No feedback on better or worse so I assume (you know what that means) it's better. Wrong!
I decide to take her car with us on a little trip today and first cold start doesn't count since I dropped the key back too fast. We get to our destination and wait maybe 20 minutes, go to leave and crank, crank, crank - no start. Turn key back and immediately hit it again and it fires up. I'm puzzled so I ask her if this is what she was complaining about a month ago and she confirms. (omitted lecture about what crank means vs. firing) We drive back towards home ~20 miles and stop for gas, no problem after letting it sit the minutes it took for gas. Next stop was longer, tried starting it in neutral after leaving the key to ON for a 10 count to prime fuel to see if something was getting in the way of it knowing it was in park - cranking with no start, tip key back and repeat - immediate start. Stopped again for 20-ish minutes, same behavior in park.
I'm a little puzzled by this - there are no codes, runs just fine when started. The only problem seems to be that first crank after sitting. I haven't checked fuel pressure yet as I can't find my gauges. At this point I'm supposing it is the fuel pump or the pressure regulator. Any other ideas come to mind for anybody else? I am split on the crankshaft sensor being the culprit but there aren't any issues when driving and no rough idle so I'm not considering it just yet. I've searched but it's hard to narrow down a no start then start issue in searches.
Thanks in advance for any other insight.