Hey there,
Short version - No, I'd say if you get the MK4 GS300h, it might be cheaper to run, but with the MK3 450h it will be difficult.
I did something similar to that:
Started with UCF20 '95 LS400 with LPG no vvti and 4 speed tranny. I got much worse fuel economy out of it but I guess the extra gear and vvti help drastically. Got it for 3 years and then went and bought a Honda Legend '06, which also had an LPG system, but is an AWD car and I think is heavier than the Ls. Sold that after 2 years of ownership and am now with a '13 F Sport MK4 Gs450h. Comparing the Ls and Gs - I'd say the Ls had a superior build quality in terms of interior and a much softer ride. The Gs feels a bit quieter though.
Still in terms of gas costs my graph would be going downwards as out of the three cars the Ls cost the least and the Gs the most in terms of gas. Disclaimer - this is in my country and here LPG is more than 2x cheaper - 55p per litre LPG vs 114p for petrol
I got a 450h knowing it will be more expensive to run, but I wanted the extra power to be there in case I need to use it. The 300h is much more fuel efficient but is slow and is only available for the Mk4.
Below are my fuel consumption numbers. Small disclaimer - I am from Bulgaria, so I am not used to the Imperial system but will give both numbers for reference. Another note highway speed limit is 87 Mph / 140 Kmh, while extra urban is 56 mph / 90 kmh here.
LS400:
City - 14- 18 mpg (16-20 l/100) LPG
Highway - about 20 mpg (14 l/100) LPG
Extra Urban - 28-29 mpg (9.7-10 l/100) LPG
Honda Legend
City - 12-14 mpg (19-23+ l/100)
Highway and Extra Urban about the same as the Ls
GS450h (4 months of owning it so far so my numbers might be inconsistent)
City - 27-31 mpg (approx 9-11 l/100)
Highway - about 28 mpg (10 l/100)
Extra Urban - estimated 33-35 mpg (8-8.5 l/100) - this might be lower, I've mostly driven city and highway.
Please note that the mk4 450h is more efficient than the mk3 despite having the same drivetrain as it supports the Atkinson cycle