Estimating discovery stages for SCRUM

Hi Guys,

I’m looking to refine my processes when it comes to handling SCRUM projects, from point of contact to start of production. I’m interested in hearing your thoughts on how best to handle the estimation / deal closure stage in particular. I’m a big fan of SCRUM, and we have been running internal projects this way successfully over the past year.

Process

1. I receive a project enquiry for a project with a decent level of complexity, I’m fairy certain that SCRUM will be the best fit, I receive a ballpark budget to work with upfront.
2. I schedule a phone meeting or casual face-to-face to discuss the detail further, we discuss the business objectives, audience, and the main user stories that relate to their most important metrics and MVP. I explain that I would prefer to use SCRUM, educate them on the process/ advantages and promise to create an initial backlog, plan for technology and price estimate.
3. I spend time researching technical solutions with the tech lead, as well as assembling an initial product backlog and estimating this with the designers and developers, I establish a per sprint cost and have an idea of how many sprints may be needed to cover the most important user stories.
4. I now have a product backlog, and a target number of sprints we would need to hit, the client is aware that given the nature of SCRUM and the project complexity, the number of sprints could vary. 
5. Client agrees and signs off on a contract which outlines the SCRUM process and client responsibilities  etc, makes deposit for the first sprint.

Questions

What if I feel that a series of discovery workshops (involves the client and designers/ developers) could be helpful to extract more detailed information from stakeholders, and possibly come up with some better ideas/ solutions? Do I include this in the estimate as exercises built into the first sprint, or would this create speculation over the initial backlog estimate?

As it will effect the backlog, do I have to sell them on doing this first, before giving them any rough sprint estimates? Seems like a tall order, "You need to spend X amount of budget on discovery exercises to discover the scope, so that we can create a SCRUM plan for you also with no fixed scope/ budget".

As some projects involve multiple days, activities and team members to get to the bottom of the problems and solutions, I guess my struggle is in the cart before the horse scenario, how do I define the backlog and gage a rough estimate for clients before these activities have taken place? Would really love to hear about other peoples processes and experiences. Thanks!