Most of our Lean knowledge comes from the work that Deming did at Toyota. There are two main ways in which I look at Lean:
- Lean Product Development
- Lean Production.
Lean Product Development is the equivalent to designing new cars, and is more like software development than production is. We're busy creating new things, not producing the same old thing again and again predictably. It's about minimizing work in progress, getting fast feedback and using those metrics to work out what to build.
Lean Production is more like a factory. The idea is to minimize variability to ensure high-quality. The only bit of software that's even vaguely like this is the build system, but even that's more like designing the factory along with the car. Particularly, Lean Six Sigma is about reducing the variability enormously, and is completely, totally inappropriate for SW dev.
The one thing they do both have in common is that minimization of work in progress. (It's a bit more complicated than this, but this is a light overview for you.)
I would be very careful, when looking at "Lean", regarding which of the techniques you apply. Fortunately SBCE is from the first, not the second, so your boss is mostly on the right page. Lean Start-Up approaches (which is where I'm guessing he read about SBCE) are also part of that group.
Lean and BDD are completely compatible; in fact, I recommend a Kanban System approach with BDD anyway. In shorthand, this translates to reducing the gap between the conversations around scenarios happening, and the actual work on those scenarios being done, while minimising the number of conversations which aren't being progressed to software (because you'll forget what you talked about over time).
SBCE is a method for running multiple concurrent experiments, which is useful when there's uncertainty around the work that needs to be done. BDD is really great at helping to work out which bits of work are uncertain, and which are well-understood or require expertise. The Specification-By-Example bit of BDD is only one part of BDD, though it's the part most people are familiar with, and that's probably why your boss is dismissing it (unnecessarily).
For more info, try this blog I wrote on using BDD as a sensemaking technique (finding those borders I just talked about), this one on estimating complexity, or watch any of my talks on BDD and Cynefin. Your boss may find the first blog I've linked reassuring as well.