I put a new bystarter in my 84 Honda Spree.. runs nice.. except after a few days of sitting idle.. the spree will only start with the assistance of starting fluid (ugh!)...
after using starter fluid -- the 'ol spree runs fine.
you might try jumping it with a car battery. don't use battery charger. car battery will turn it over a lot faster and maybe better spark. your spree battery may not be up to spec.
i hate to tell this embarassing story.. but..
i tried to start mine, which ran a day prior.. and i was like *? , so i took the carb off, changed the plug, which looked brand new, and flushed the tanks.... i tried to start it again and nothing... WHAT IS UP!?.... then..
If it only starts with starting fluid it sounds like your bystarter that you replaced is not working. Is the o ring good on your bystarter? If it starts good after it is warmed up it sounds like it isn't choking properly when it is cold which is what your bystarter does. Are the wires hooked up good to your bystarter?