Yes, the LSP supports unannounced/private channels. If you set announce_channel: false in the channel request, the LSP will open an unannounced channel with you.
{ "node_connection_info": "string", "lsp_balance_sats": 50000000, "client_balance_sats": 0, "confirms_within_blocks": 6, "channel_expiry_blocks": 12288, >> "announce_channel": false }