Engineering is meant to be very practical; the polar opposite of an academic education. You need the academics to give you a foundation you can work with, but the actual practice is something you can only get with experience.
The start of that experience is the Engineer In Training (EIT) exam. If you pass that, you're deemed to be fit to work as an engineer under the tutelage of someone with full PE certificate. However, the EIT has virtually no merit among academia that I'm aware of.
Please understand that there is more to Engineering than just academics. Even those with engineering educations are completely clueless when they get to the work-place. Designs can be done many ways and there are many reasons for choosing one design over another. Understanding the implicit requirements, not just the stated requirements, understanding the safety and security trade-offs of each design, and understanding what the client actually needs is crucial. The main reason those with engineering education backgrounds do better is because they have actually been exposed to a few methodologies and shortcuts that a physics student might not be aware of. However, they have no experience selecting one design over another. For that, the only solution is to get experience.
I'm suggesting to you that by studying for and taking the EIT, you can avoid the school part and go straight for the experience. Then, after you have the experience, you can consider whether to further your education or not.