Your experience of reduced flow when using mains water is not uncommon. The type of pressure reducer valve used in teh hose pipe may simply be unable to match the flow from your pump, but also the low flow may be nothing to do with your equipment, it could be poor flow at the sites tap.
All of the systems I have seen the mains water pipe replaced the pump, so its is simply impossible to use your pump to assist teh flow.
However there is a way round it. You can get kits that allow you to fit a torbeck valve to a water container. This allows you to use your water pump, but the water container is automatically kept topped by the mains supply.
It also adds another level of safety, as if for any reason the mains pressure reducing valve fails ( Which by the way is a very rare occurrence), then the mains flow will simply over flow teh exterior water container and not flood the caravan.