My main belief that the 2nd restoration happened after Cromwell died was that after the 1st civil war, everything seemed to fall apart: famines, army dissatisfaction with parliament due to lack of pay, and Charles plotting with the Scots to invade England to place him back on the throne. All of that I believed prompted people to long for the days of stability and order, which they had under a king.
So if we can just slow everything down a bit to a more reasonable pace where English society doesn’t breakdown. I think the commonwealth can survive.
Step1: parliment not passing the self-denying ordinance, which says you cannot serve in parliament and the army at the same time.
Step2: after 1st civil war is won, parliament exiles Charles, thereby preventing the 2nd civil war.
Step3: reform parliament by allowing universal male suffrage (males only, sorry ladies but women voting in the 1600's is not possible no matter how progressive the society is)
Step4: reform house of lords-keep intact but they cannot vote, but they can submit proposals to the commons for passage.
Step5: purge more radical elements from New Model Army/keep up payments to army.
Step 6:religious reform-all religions will be tolerated as long as they are not subversive to the commonwealth. Tone down puritan rhetoric.
Step7: abolish institution of monarchy.
It is assumed that if the people of England have peace, security/stability and prosperity under the commonwealth, they will not have to look back upon the monarchy as a time of nostalgia because they already have everything they need to live.
Now if all of this could happen, then getting rid of the institution of the monarchy should not be a problem because if there is peace and people are happy they will hopefully have forgotten about an old and outdated institution that no longer has any relevance to their lives.