The clear answer is..... yes you can get a single impeller pump to supply additional pressure to just one water utility.
The reason why its not advised, or works in the best formate, is one should really only supply mixer showers with balanced pressures.... ie either both low pressure or both mains pressure (or pressure reduced).
Therefore, for best performance etc, you need a dual pump device pumping both low pressure hot and cold.
It means more plumbing in essence, which a lot of end users choose not to have done, (or undertake themselves), as the means of getting low pressure cold water is tricky.
Low pressure cold (or tanked cold) is the water that feeds your hot storage cyinder tank.
You may need to breach into the coldp storage tank ( its either above the hot cylinder or presumably in your loft), with a new 'take off'.
Make sure (if doing such a task), you get a take off point slightly higher than the bottom of the tank base, (say 50-100mm up), to avoid taking sediment from the base of the tank.
Sedimentation will give you a 'dirty' shower, block filters in the pump and/or shower, and reduce the life of the pump (also known as Booster Pump).
I would consider a Combination boiler before undertaking this work, unless your means of getting hot water is only via electric immers immersion heating.
Edit:
Single impeller booster pumps are also very noisy.