Unfortunately, it looks like the General and Custom shipping profiles that allow you to apply different rules for various shipping classes is independent of the section for processing time, (which seems to be a global setting that applies to all products on a site).
Unless there is something I’m missing, a better solution for Shopify might be to move the processing time settings inside the general and custom shipping rate profiles so items assigned to these profiles can not only have different shipping rate classes, but, also different processing times.
If any items are purchased across profiles with different processing times, the rule would then be to calculate a single shipping estimate based on the profile with the longer lead-time, or, allow the customer the option to opt for two different shipping methods and delivery times, (as well as the applicable shipping costs for doing so), prior to payment. This is similar to what Amazon does when you buy different items from different sellers with varying shipping rates and delivery timeframes within the same checkout instance.
Until they add a feature like this, the only thing I can think of is to turn off the option that shows the expected delivery dates at checkout and put up a banner or some kind of announcement in the cart that says all non personalized items ship the next business day while orders with personalized items ship in 2-5 business days.
Per the example below, I personally also add the expected shipping time for each product right below the price to set delivery expectations prior to checkout and incentivize engagement:
https://www.mckinneyprintingco.com/products/highland-cow-ornament-custom-custom-gift
In this case you can have a product template for personalized and non-personalized items along with the banner in the cart that clearly sets the expectations for shipping.