I think it's hard to be widely successful in any public sphere without being able to keep a verbal distance between your faith and your Recorded statements. Santorum crosses that line continually and so is almost certain to be unsuccessful as a presidential candidate. Romney would be unelectable if he ever appeared in public wearing his temple robes or spoke even remotely in defence of a Mormon specific policy ( like intervening to get missionary access to a country in return for foreign aid ) and rightly so. I think to see how his religion influences his public life you need to look at his previous jobs.
What I see is the archetypal Mormon stake president - stake presidents are generally elected ( cough chosen by the spirit) because they are negotiators, fiscally sound, have a solid traditional family relationship, loyal to authority, not boat rockers, good listeners and father like. As a SP you need to sit through the trials and tribulations of congregation members as they come to confessional style meetings where the SP is guided to balance mercy and judgement with the overall aim of getting individuals to fix their spiritual and physical lives and become mentally and materially self sufficient. As such Mitt will be quite in touch with the arguments put forward by different sectors of society ( albeit working class, middle class and cash class Mormons.) in short Mitt's strength will never be on the stage it will be in the post address strategy meeting. This is a problem for him as President - the calm, negotiator stle leadership that serves well in mormondom isn't necessarily what is useful for leading a nation where a bit of fire and inspirational chutzpah are often critical ( Churchill, Napoleon, Reagan etc).
Should Romney get elected, other than a hardening of the 'gays destroy marriage' tripe, it's hard to see what part of current Mormon theology will affect his stances given that the church heirarchy is currently scrambling to erase its socially crippling past and present a mainstream, apple-pie religion. Mormonism has a laid back cockiness about itself in stark contrast to the more sensationalist evangelical movements or the doom laden JWs. Mormonism isn't overly obsessed about the second coming or in putting in place rapture/ Armageddon scenarios partially because it put in place extra unachievable steps as signs of the times ( like building an actual Zion in Missouri) so with those mental buffers they don't need to sweat about the other nastier signs of the times, I.e. the second coming isn't just around the corner.
While many may mock the concept of becoming a god it has one very positive side effect. Mormon teachings see this life as being a place to self improve and get ready for great responsibility by self sufficiency, education and life success. These fit quite well with core republican aims. Mormonism also has quite a practical side to it involving preparation ( food storage etc) and charity ( Welfare Square ) which is partially why Romney wouldn't have struggled to pass Romneycare and could live with Obamacare ( it's not fundamentally a bad program - its just not a clearly costed program.) Mormonism is strong on self sufficiency and always tries to tie charity to enabling the recipient to pay back ( so a family getting its groceries paid will be expected to go clean the church for a few hours each week etc ) however, the church also will give untied aid when the need is urgent rather than long term. Mormonism is a good synthesis of Republican' work for what you get' and the Democrats 'compassionate state' approaches. In fact historical Mormonism has closer roots to Communism than many might know ( see United Order ) even though recent church leaders have shown near hatred of Russian Communism.
Lets finally look at the most Mormon views that could cause issues for a Mormon president:
1/ marriage - Mormonism cannot allow Romney to champion legalising gay marriage since it has tied itself to that mast. Gay marriage is also seen as a gateway to polygamous marriage the legalisation of which could kill the Mormon church from within.
2/ American exceptionalism. - Mormons really believe the New Jerusalem will be built in the US. America is part of Gods divine plan ergo it's rights must always be placed over other countries. Note Mitt's hard line stance against Russia and China. He will fight passionately to ensure America wins in trade and all international exchanges. Might make a Mitt more extreme than an Obama.
3/ social provision - expect to see this cut. Mormonism ultimately hates to waste money unless it can see a reciprocal 'improvement' in the individual. There is little tolerance for free loading or laziness. Expect a Mitt to push workfare over welfare.
4/ education - apart from ID which Mitt would potentially push but not as much as Santorum the educational system would be in good hands. Mormons like education.
5/ the biggie. Will the Mormon church try and influence public policy. Publically definitely no ( political and social suicide for Mitt and the church ) but privately - hmmmm - if you mix socially on a regular basis with Mormon aristocracy how can you not be having an influence? The most powerful person in the world will now be a phone call or an interview away from the Mormon prophet. Whatever Mitt says publically , for a true Mormon, the prophet trumps the public ( to be fair that has to be a ditto for a strong Catholic and the pope.) It is not inconceivable that a Mormon prophet could demand certain things ( Either through illness or an inflated sense of moment - and we have some nasty, arrogant bar stewards in the Mormon heirarchy) that would cause Mitt to need to either resign or do something stupid. What does a Mormon president do in a nuclear crisis if the prophet says god wants a preemptive strike? Not a good pressure in a hot moment. When it comes to government you can't afford to have a potential unelected loose cannon influencing the leader no matter how unlikely that is. Mormons truly believe they are the only true church and that they really are being led by direct revelation from heaven. Imagine some of our more convinced true believers on this forum being in charge of the country. Maybe very lovely 99% of the time but utterly capable of getting a 'message' at any time and ignoring all evidence basing action upon it.
Quite unsettling.
Personally,I'd stick with Obama and get Mitt into a financial management role.
