A variation of 5 meters of head in a centrifugal pump discharge line is not of much consequence in a real world application. The flow control valve in a real pump system will likely have at least this much differential pressure drop range. The pump should be sized for the normal operating conditions and the system checked for operability at all expected operating conditions. The difference in height of 5 meters to the highest point does not matter in itself... it is the variation in total system differential pressure that matters. This will no longer be a 2 to 1 pressure range as in the simplified assumptions. The system pressure drop calculations should take into account the line, check valves, control valves, and the other piping devices. When the flow control valve and the entire pump system are sized properly, a single speed centrifugal pump should be a fine choice. Pump systems typically see many more operating conditions than a single design operating point.
Theoretical questions with sparse description are an okay diversion for discussion but I learn more from the real world problems that appear in these forums. I consider possible 'what if' objections to my first paragraph. What if the pump and system are very small? Then efficiency won't matter. Just size the pump as needed. What if the pump and system are extremely large? Just size the pump as needed. If 5 meters of extra differential head on startup make the system inoperable, then adjust the system so that it can operate. A VFD drive might make sense for a very large pump, but there would need to be more justification for the cost than an extra 5 meters head on startup.