I've experienced ecommerce sites where after adding an item, a modal pops up that gives me the option to checkout or continue shopping. It's not entirely annoying since I typically don't add a ton of items but can imagine how it could be. Dictating a user's experience for them (i.e., forcing checkout) is a surefire way to get people to abandon.
You could make the CTA more aligned with the action that is going to happen next such as not labeling it as "Add" since you are technically not adding it to a cart or bag, but maybe label it "Buy" instead so it makes it more clear that you may head towards checkout by clicking on a "Buy" button. Additionally, if the client is wanting customers to buy more even though checkout is forced upon them, on the subsequent page, you could have a list of recommended items or "customers also bought" with a quick add button.
When you click on the [Checkout] button, does it show the items first? If so, maybe you could still get away with labeling the checkout button as [Cart]. I'm a proponent of having a dedicated cart, but I also know the hassle of stakeholders not budging or willing to allocate resources to dev so I'm trying to think of how you could create the illusion of a cart to the customer with this out of the box 'solution.'
Lastly, like someone else said, grab a few users and have them go through your store. Try to record their responses through video or audio to show your stakeholders if they can get through the experience without a cart. You could even send out a questionnaire with questions like, "Can you explain step by step what you believe a retail experience should be like?" "Where would you go to change the quantity of an item?" etc.