I think the reliability of charity in this case would increase as the number of people who need it decrease. I don't think we can rely on charity to solve everyone's financial woes, else said woes would already be solved. But when the percentage of people in need of charity grows smaller, the impact of the same pool of charity increases for those left in need.
But isn't the point of BI that everyone has enough for their basic needs? It would seem unfair at that point to tell some people try and get charity funding (which might dry up at any point).
There are plenty of people whose basic needs are substantially larger than the average citizen. For instance, many people need expensive motorized wheel chairs, which would be difficult to purchase on BI. Currently, the state provides motorized wheel chairs to many, but there are many stories of this program being abused. It's quite possible that a charity could step in here to supplement BI to make chairs affordable, and could do better and more efficient needs testing than the state.
According to Maslow's Hierarchy, at least as I interpret it, charity increases as an individual is able to meet their lower needs. How you go about getting more individuals to that level is a matter of debate that is difficult to give any definitive answers to.