Very unlikely. Changing the constitution requires a two thirds majority and he was never going to get that popular. The Socialists certainly did not want a restoration of the monarchy and after he threw them out of power, the right-wing UDF didn't want it either.
Of course it's another question how he could do better. As
this article points out, he was defeated despite four years of sustained economic growth and securing EU membership. But the massive corruption and slow improvement in living standards destroyed his popularity, and the decision to ally with the extremely unpopular ethnic Turkish MRF didn't help much either. Perhaps more importantly, he made an extravagant promise that his government would cause noticeable improvement in living standards in 800 days and this failure seriously damaged his credibility. Avoiding it might be a good idea. Other problems are more difficult to fix. Due to tight requirements to enter the EU and a restrictive currency board, there was little that could be done by the state to improve living conditions. Reducing corruption would require both a better selection and control of the people in his party (difficult, since most joined in hopes of profiting from corruption) and of his coalition partner (even more difficult). Perhaps winning a majority would help there, since at least there would be fewer positions that would need to be filled with supporters of the government (not to mention avoiding the hated MRF). And of course, not stealing properties that didn't belong to him would definitely help his popularity (but then again, that was probably the whole point of his return to Bulgaria).
Well, many think Simeon II ruined his monarchist chances when, becoming Prime Minister, solemnly swore loyalty to Bulgarian Republic, as the Constitution required. When his not brilliant performance caused him to lost first position in 2005 election, he formed a coalition with left parties that of course destroyed any monarchist revival. If you want Simeon II retained his Crown, he could avoid a direct political role, appointing someone other as Prime Minister (possibly a competent person) and obtaining a referendum.
Such a government would be a joke. A puppet prime minister who is not independent, yet is still responsible for the actual government, while there is no accountability for the actual ruler. Saxkoburgotski tried to do this in OTL and was quickly persuaded that this position was not tenable.
Also a referendum is not a constitutionally mandated way to change the constitution.
Entering to politics was already big mistake for any hopes to restore any kind of monarchy. Him would had much better chances if he would had remained outside of politics and just supported restoration of monarchy. In Europe royalties never should participate to politics but them should be above of that.
Or alternatively, he could have run for President, which is a less political office and would allow him to become an unifying figure for the nation. But the Constitutional court ruled (narrowly, 7-5) that he didn't meet the eligibility requirements. So there you have a possible POD.