Undergrad thesis supervisor rejected request for LOR - how will this look on grad apps?
It's generally a good idea to have at least one reference from a research supervisor. It's sounds like you have one, possibly two, such references, so that is no issue. Given that you've only worked with your current supervisor for a few months (and haven't even submitted your proposal yet), I don't think the lack of a letter from her will be suspicious. 

In terms of the options you list:


As far as I know, getting extensions on grad school apps is not really done. 
You could wait until the next cycle, but that seems extreme. 
Rather, I would just proceed with your application as-is, clearly indicating that your primary research mentor was [the person who wrote your letter] and that you only recently started working with [the person who didn't].