I have a few questions about Location Push Service and Screentime Family controls.

To be able to use both APIs even in your test environment, the required entitlements must be assigned to your Team ID first. In your words, you need the permissions for both.

Whether your application will be approved, how long it will take, and so on are not questions we can answer here. The entitlement teams will take the necessary time to process your request and make a decision.

What I would suggest is to read the requirements for the entitlement requests very carefully to not waste time waiting for a request which will end up being rejected.

Entitlement requests will be carefully considered, and will only given for specific uses as described. For example, if you are expecting to get the location push entitlement so you can use those to control Family Controls settings, your chances for a successful entitlement request will not be high.

To answer your question, the Location Push extension is specifically meant to be used to get the user's location and share it with other servers. Other uses may not be permitted - but in the end this will be up to the entitlements team.

Because using Screen Time APIs in a Location Push Extension is an off-label use, we can't give an answer one way or another as it is not a supported use case. To test the concept, you can always use a Notification Service Extension without entitlements to see if what you want to do is going to work from a push extension.

Although, a location push extension may have different sandbox rules that restricts access to various system resources. It is not possible to give a hypothetical answer because a lot will depend on what APIs will you be using exactly, and what system resources you will need to access to accomplish your task.