We don't document the Kibana URL structure/format, and we are planning to make some changes to it, but in the short term, you should be able to specify the time range relatively easily. To see how, change the date range to absolute in your dashboard view, and observe the URL that is created. It will have something like
?_g=(refreshInterval:(display:Off,pause:!f,section:0,value:0),time:(from:'2016-02-10T16:00:28.681Z',mode:absolute,to:'2016-08-11T15:00:28.681Z')) somewhere in the URL itself. Your Watch could use templating to replace those dates.
See here: Access discover tab with parameters for a broader example of applying filters.
Again, a word of caution - we're planning to change the URL format in future versions, so this won't work forever and isn't something the Kibana team can directly support. Having things like this customizable via URL is something we'd like to maintain with future approaches, but the details of what's coming are TBD. See: https://github.com/elastic/kibana/issues/4338 for some context!