It's not uncommon to get a client request like "I want the design to help decrease bounce rate" or "increase conversion." What's your experience with designing to help increase those metrics?
In my experience, the first thing you need is data. Look at the data on Google Analytics to find out where you are right now and identify where the client wants to be in x months.
Then use recording tools like Hotjar to observe how people use the website. Evaluate whether the business goals are aligned with the user's goals. It won't help conversion if the website's goal is to sell super-premium organic quinoa but people come for their blog articles titled "Healthy Dinners Under $5".
If the goals are aligned, find out what deters the users from converting. Simply watching a few video recordings on Hotjar will give you good hypothesis on the user's pain points. You may notice that a user couldn't find the add to cart button when they scroll up and down their mobile screens several times on the Product Page. Or they abandon cart after pausing for a few seconds at the shipping fees. Or the website make them jump through hoops when checking out. Worse, pop-ups.
One thing I might add is look at the metrics as a puzzle with many layers. I find it helpful to ask client questions like "Would you like to increase conversion for a specific goal (i.e. Product A) or across the board?" That can lead to interesting answers (example: high conversion on Product A will yield much higher profit margin and position the brand better than small conversion increase for all products).
Hope that helps!
Lots of good thoughts here, thanks for laying out what your approach (starting from data) would be.