The family of a sick toddler whose illness helped persuade the New Jersey governor, Chris Christie, to change his stance on medical marijuana for children have said they will still have to leave the state to save their daughter's life.

Christie announced on Friday that he planned to lift restrictions on certain forms of the medication, including the high-dose edible marijuana that is needed by Vivian Wilson, a two-year-old fighting Dravet syndrome, a severe form of epilepsy. The girl's father confronted Christie at a campaign stop on Wednesday, to complain that the governor's office was dragging its heels over a decision on a more restrictive bill that was introduced to the state legislature in May.

Brian Wilson said he welcomed Christie's policy shift to conditionally veto the bill, but that the months before the changes would take effect was time Vivian did not have. "We can't wait a year before we start treating her," Wilson told the North Jersey Record.

He said he felt it was "a selfish victory" because Christie appeared to be reacting to his family's situation, and he was also critical of restriction the governor left in the bill that would require the approval of both a psychiatrist and a physician registered by a state medical marijuana panel before any treatment could be made.

"It makes a lot of headache and heartache for parents to go shopping for doctors who understand anything about medical marijuana to get them to sign up for this," Wilson said. "For parents who are already going through a lot of trouble just with what their children's ailments are, they now have to go through this extra stuff you don't have to go through for any other medical condition or for any other medication."

Wilson and about 100 supporters effectively ambushed Christie during a rally at a restaurant in the family's home town of Scotch Plains. The governor appeared flummoxed as Wilson pleaded with him: "Don't let my daughter die."

New Jersey governor Chris Christie. Photograph: Jeff Zelevansky/Getty Images

Christie had earlier said he was concerned that the easing of restrictions was "going down a slippery slope" and that it would make it easier for medical pot to fall into the wrong hands. On Friday he said he would approve an amended form of the bill if it stipulated that edible marijuana, in tablet or lozenge form, would be available only to minors.

"I believe that the parents, and not the government regulators, are best suited to decide how to care for their children," Christie said. "Protection of our children remains my utmost concern, and my heart goes out to those children and their families who are suffering with serious illness."

Vivian Wilson's condition requires an oil-based strain of the drug, currently banned in New Jersey – medication high in the seizure-reducing component cannabidiol (CBD) and lower in tetrahydrocannabinol (THC), the active ingredient that gives users a high. On some days she suffers up to 100 seizures, said her mother, Meghan Wilson. Brian Wilson said Christie's veto was "a step in the right direction".

"There's a lot more parents in New Jersey who are going to have to get into this and they're going to go through the same problems we did," Wilson told the Record. "The decision works for us. It just could have been a lot better."