How can I get all my gas LPG radiators and hot water to work?

I recently moved into a house with an LPG heating system (it was installed in 1994) where the heating and hot water are not working properly. With the exception of the bathroom and kitchen, the radiators are antique cast iron. There is a total of 16 radiators in the house. The system has been flushed in the past when a similar problem occurred. The previous owner was under the impression that the boiler is unable to heat all the radiator and hot water, so recommended I use the immersion to heat the water in the winter . Initially, I followed the previous owner's advice (the heating hadn't been used since last winter), but still three of the radiators (one modern one in the kitchen and two antique ones in the bedrooms) didn't work. Two of the non-working radiators were completely cold (one upstairs and one downstairs), the other was warm at the bottom only. I called an engineer in, who checked the valves and boiler which all seem fine and he also checked there are no air locks in the system. Even so, he couldn't get them to work. He told me that the boiler (a Potterton 100E fanned balanced boiler) is plenty big enough. The previous owner said he looked into buying a bigger LPG boiler but there wasn't such a thing.
Following the engineer's visit, I then tried to heat the water and radiators using the boiler. Then, the water didn't heat up and three additional radiators stopped working (one in the bedroom and two downstairs), plus one radiator started giving out far less heat. I called the engineer to explain what had happened, but he didn't seem confident in diagnosing the cause of the problem and suggested a flush as a next step. However, I am not convinced this would solve the problem. There are two Grundos pumps fitted on the system (one next to the boiler, in the garage, 15-50) and one in the airing cupboard, in the house, 15-60), which is something that the engineer had not come across before. There is also a Monsoon 1.5 bar pump fitted to the hot water tank. Any advice would be greatly appreciated.