I do wonder f there is a important difference between "powered by impulse," and possessing "impulse power."
Just a impulse engine?
A warp drive powered by impulse?
A valid idea. The matter/anti-matter reactor which provides power for Star Fleet vessels is typically referred to as the 'warp core'. Perhaps Romulan ships of this era did not have such a power source, instead relying on less powerful and efficient substitutes? The fusion reactors that power the impulse drive, perhaps.
Indeed, if the warp engines can
be powered from the fusion systems, this would explain how the damaged Enterprise could travel to Delta Vega in 'Where No Man Has Gone Before'.