How do you know it needs 5000 BTU? Did you have a heat load performed?
If it does need 5000 BTU and you are looking for a ductless solution then a 9000 BTU would do nicely. The Mitsubishi M series uses VRF technology to modulate output capacity. IOW it modulates the speed of the compressor, outdoor fan and indoor fan to match the load of the room. It will modulate down to 20% of total capacity and the H2 will do a little better than that. Mitsubishi is made in Japan.
LG has some models that will modulate as well. Made in Korea.
Fujitsu has some modulating models and is made in Japan.
Mitsubishi has the largest support network for parts and systems, Daiken is next, then LG then Fujitsu; in my area anyway.
The Japan products are superior, I am a Mitsubishi dealer BTW, and parts are relatively quick to find. Korean and China systems are a nightmare and parts come from a china on a S L O W boat.