I'm thinking of having something in Classic that both switches to Lightning and starts in a specific Lightning App. Perhaps all that is needed is an official, documented API that does the equivalent of what the "Switch to Lightning Experience" link does:
<a href="javascript:sfdcPage.handleSwitchToLightningClick(%27LightningExperiencePreferred%27,true,%27/ltng/switcher?destination=lex-campaign%27);" class="switch-to-lightning"><svg>...</svg>Switch to Lightning Experience</a>
with the Lightning App pre-selected as the default.
Is a URL link sufficient to switch e.g. from Classic:
https://java-drive-8200-dev-ed.cs18.my.salesforce.com
to Lightning:
https://java-drive-8200-dev-ed.lightning.force.com