Late 2008 Macbook Pro will not start if left unplugged overnight

Question

Hello,
I've been having the following problem on a Late 2008 MBP, only owner and this just started a couple weeks ago (along the cold weather perhaps?):
Every morning it's very hard to start the MBP, I press the power button and get nothing, not even LED light. I already did SMC & PRAM reset with no apparent improvement on the issue. This happens even when the battery is fully charged, which was replaced for a new one two years ago.
I manage to start the machine after several combinations of removing the battery, plugging the AC adapter in it, butting the battery back in, etc. yet I haven't found a way that actually works.
If the machine is left plugged to the AC adapter overnight, it will boot without any hassle the next day! Somehow is leaving it on battery only that bricks the thing.
Is this a problem with the motherboard? Once it starts I can turn it on, off, send it to sleep and be on battery like the machine is brand new! HELP!
OS: 10.9.5
AC Adapter: 85W replaced around 2010
Battery: Health 96%, Cycles 143 (funny, yesterday was 97% and not long ago 99% or even 100%)
Additional peripherals: Expresscard USB3 (has been removed for testing)
Mods: 8GB RAM & Samsung 256GB SSD

Answer 1

Thanks for taking time to respond to my post, I understand it's old hardware but hey, if there are Macintoshes still running in the wild, why can't I have this MBP run for another decade!
I did what you suggested; cloned the SSD to the original HDD and replaced the RAM with the old Samsung memory – got the exact same result.
In fact it was harder for me to start the machine this morning. I couldn't do it with the old HDD and RAM so replaced them with the new ones AND still struggled to boot. I even reviewed all connectors where properly in place within the motherboard, no dust or anything strange inside the machine. I removed and re-inserted the RAM, did SMC reset and that seemed to do the trick (this time the machine booted only with the battery inside).
I read somewhere there's a thermal sensor involved in the booting process, could this be temperature related? Maybe moving things around just happens to warm up the parts. Are there any other type of tests I can perform on the RAM/Power Management to rule out any other problems? Is there an internal battery that might need to be replaced? I haven't suffered from clock reset or anything like that. I've had other computers not boot at all, both Mac & PC, and it's usually GPU related, this machine doesn't behave like that at all but could that be the problem?
Like I said, if I keep the machine in sleep mode or plug it in to charge before turning it off, there's no problem booting the morning after. Once the machine starts, I can turn it off and on even if the battery is removed and have never experienced a malfunction while on. It seems the lack of power is what bricks the machine regardless of the SSD/RAM or peripherals connected to it. Battery is charging to 100% again.
Any other suggestions are welcome! Will try not to leave the machine off or disconnected in the meantime.
Thanks again!