Back in the day, when Ford released it's CVH powered Sierra, it was great for a while. But then they seemed to be going through starter motors like a kid goes through nappies.
But, somehow they didn't relate the fact that they were also going through batteries too. The car would become harder & harder to start, so a new battery would be fitted and all would be well....for a while....
Anyway, it turned out that the timing was being adjusted electronically and would advance, but "only" during cranking! This made it really hard for the starter to turn the engine over, ergo the battery would struggle too.
Now my point being, you have fitted a replacement battery, but even though it is marginally better, there seems to still be a struggle going on.
I would be checking the wiring from the battery to the fuse box/alternator and so on, especially, as previously suggested, the earth points.
Have the guy check the resistance from the battery+ to the fuse box and all the earth points. Sounds awfully like a high resistance connection some place under the bonnet