I can't help but feel a contradiction between the notion of being a bigger fish in this union while at the same time having a "puppet government".
On the point of the euro, the Treaty of Maastricht requires states joining the EU to also join the eurozone once they meet the necessary prerequisites. The actual meeting of those prerequisites is a different matter, which is how Sweden has been studiously managing to avoid adopting the euro for twenty-six years now by simply not participating in the European Exchange Rate Mechanism
There would be need a need for a customs border, in the least, if Scotland joins the EU's customs area. It would need to operate in the same way as the customs border between the UK and France, unless there is an agreement for deviation towards something like the NI Protocol.
An immigration border would be necessary if Scotland were to join the Schengen Area, as is required of new members of the EU - but it could potentially be avoided by a hypothetical special agreement that kept Scotland in the Common Travel Area instead.
But yes, the details of this sort of thing are a bit of a drift for this particular thread.
I doubt the CTA in its current form would survive the break up of the UK. With customs borders there is little benefit of passport less travel and there are security benefits of having passport controls. I think the primary benefit of the CTA (FOM) would survive the break up of the UK but there would be passport controls on the Tweed and across the Irish sea. Wales would be a mess if it went independent, a whole other topic.
Your technically right about the Euro but your missing the need for Scotland to convince EU members that it will join the Euro. UK remainers don't seem to understand that to much of Europe the EU and the Euro are the same thing. One pro Euro country will veto Scotland's application if it doesn't think Scotland is sincere about joining the Euro as soon as it realistically can.