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However you're generating an invoice, there should be a route hint included in the result. This tells the payer how to get sats to your node if it will need to use private channels.
Try generating an invoice and decode it. If a route hint is not included for your private channel(s) with inbound liquidity, it could be impossible for a sender to complete the payment. Generally your node should know that payment will be impossible without a route hint (if that's indeed the case) and it should include one without prompting.
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Here's a lightning invoice decoder
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This is an invoice of mine, I also have a private channel that's receiving sats from SN just fine.
The Public Key and short channel ID are the routing hints, correct?
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Yes, here it's telling the sender to get to the provided node, then ask it to use that channel. The channel is using an alias, because it's not public, but that node will know which one you're talking about. Then, the sender can compute the route and factor in how much this peer charges to use this channel.
Your node can add multiple of these hints and each hint can even have multiple hops. It should be smart enough to figure out which hints it needs depending on the size of the payment and how much inbound liquidity you need.
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yes, the routing info node should be your LSP nodeID, you can check it in any LN explorer
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