There seem to be some misunderstandings here.

You are probably right, I don;t have the battery in front of me to
check.

Firstly there is no such a thing as a 4.5A battery, what you have is a
4.5Ah battery. This tells nothing about what current it will supply,
but rather its capacity.

2nd, you wont get 1.5hrs from a 4.5Ah battery on 3A, for 2 reasons:
a) Ah capacities are only correct for 20 hour discharge. High
discharge rates will reduce capacity considerably.
b) If you discharge a lead acid battery until flat it wont last long.
You need to stay above 20% charge at least.

For 4 hours of run time, if we momentarily ignore the 2 above issues:
4x3 = 12Ah
Add 50% (guessed) for a short run time - 18Ah
Add 25% to permit discharge down to 20%: 22.5Ah

So really you want a 25Ah or more battery to do what you want.

that is probably a good start.

There is one last issue. Is the monitor 12 real volts or 12 battery
volts? What are known as 12v batteries arent really 12v. This is fine
if the monitor is designed to run off car supplies, but if it needs a
12v regulated supply it might be unfine - or it might be ok. Would be
wisest to check first.

NT

How do I check? The monitor did not come with instructions and the
specs I am detailing are from the back of the monitor, so I cannot
really tell the voltage it needs. IS there a way to check this
reliably?

Thank you for the interest.