I'm not sure when this feature would be available in Shopify shipping. But, if you are open to using an app for this, the Multi Carrier Shipping Label app would solve your problems. With the box packing method in the app, you can set up different packages for your products based on the weight & dimensions.
So, whenever your customer places an order, the app automatically chooses the right box and calculates the accurate carrier calculated rates accordingly relieving you from being overcharged.
This is Finley from the Social Care team at Shopify.
I understand that it would be ideal to assign specific packages to specific products.
What I can do is add your name to the list for this feature request.
Our development team will use these requests to see which features are most required to be built into the platform.
I've sent you an email to your Shopify Community email address. You can respond to that email and I can add your name to the request.
In the meantime, if you're offering Shopify Shipping's calculated rates at the checkout, the amount that your customers pay for shipping will be determined by the weight of your product(s), the dimensional weight of your default package, and the shipping carrier selected.
You can update the dimensional weight under your Shopify Admin > Settings > Shipping and delivery > Packages > Saved packages and then click Edit beside the package with the Default label.
However, if you've already adjusted the default package to the best size and weight you can, there are a couple of other options you can look into:
1) You can add a handling fee to mark up the calculated shipping rate to help cover additional costs.
You can mark the shipping rate up by a percentage of the total shipping rate, or with an additional fee.
Your customer would see this as one unified shipping rate when they reach the shipping page of the checkout.
You can edit this handling fee by going to your Shopify Admin > Settings > Shipping and delivery. From there, click Manage rates next to the shipping profile you'd like to edit. Click the "..." next to the rate you'd like to adjust and then click Edit rate.
You'll see the Handling fee section where you can adjust it like so:
Make sure to save your changes, and you'll be good to go.
2) Apps from the Shopify App Store allow you to have more flexibility outside of the one default package.
A couple of apps on the Shopify app store that you can have a look at, alongside the app mentioned in the post above are:
Thank you for your understanding. If you have any questions for me, please do let me know.