Well going through the Danish history:
1743 prince Frederick is married to princess Louise, daughter of George 2.
1746 king Christian 6. dies and prince Frederick 5. is now king. A 15 - year defense treaty is signed with Russia and France guarentees the Danish possesion of gottorp Sleswig. Empress Elizabeth won't give a guarentee but declares her willingness to find a peacefull solution to the Sleswig question.
1750 Swedish pretender Adolph Frederick renounces his claim to gottorp Sleswig and should russian pretender Charles Peter Ulrich die heirless making Adolph Frederick heir to gottorp Holstein, Adolph Frederick promises, under swedish government pressure, to an exchange of property, gottorp Holstein for Oldenburg-Delmenhorst.
1751 Danish minister in Russia get Katharina, later empress, sympathy for the 1750 agreement. Queen Louise dies.
1756-1762 The Seven Years war. Denmark is kept out of conflict by foreing minister Bernstorff in order to reap the benefits of neutral shipping.
1762 Empress Elizabeth dies and Charles Peter Ulrich is the new zar, Peter 3. Peace is made with Preussia and threats issued against Denmark. Denmark mobilize an army of 70000 and the navy is sent into the Baltic to await the assault. Count Saint-Germain move into Mecklenburg with 30000 men taking up position east of Wismar. But war is avoided by the coup of Katharina 2. on July 8.
So you would have to have a settlement of the Sleswig - Holstein question involving the pretenders of both Sweden and Russia. Other actors in this question was France and empresses Elizabeth and later Katharina. Without this settlement, Denmark would be keen to avoid being dragged into a war against Sweden but especially Russia. Going to war in 1756 against Sweden would mean to go to war against Russia and France and no settlement of the Sleswig - Holstein question, which was essential for Denmark at the time. That would overshadow the benefits of an alliance with Britain, that nobody in Denmark at that time would have any indication of would be of importance 45 years later!
Having a independent or semi-independent hostile duchy on the land border with Germany had been a bad experience for Denmark, that had to be dealt with. Russia, Sweden and France were willing to settle the problem.
The question was settled with Russia in 1773.
The right of passage for neutral ships was a Danish priority, made by Bernstorff, at this time and again in 1780 and 1793. (Well not Bernstorff at these later instances!)
Then get the Sleswig question settled, get Danish foreign minister Bernstorff out of the way, get the danish king sober and keep queen Louise alive for at least 5 more years... then you might get it your way.