Why not reverse the question by finding suitable husbands?
Anna, widowed duchess of Courland, marries illegitimate son of King August II,
Hermann Moritz Graf von Sachsen (or in French
Maurice de Saxe). In OTL the marriage fall apart by 2 reasons: 1st, Catherine I was afraid that this would strengthen the Polish position in the Duchy (or perhaps somebody told her so because she was not exactly the brightest apple on any tree) and 2nd, Menshikov decided that it will be a good idea to became a ruler of the (semi-)independent state and screwed things up to a degree that required a prolonged and not 100% successful patching of the situation.
Maurice was liked by a local nobility (of
both genders; it seems that within a short stay in the duchy he managed to became very familiar with a considerable number of the local females) and Anne liked him as well. So let's assume that Catherine does not object and keeps "Danilovich" at bay (difficult but quite possible: he was recalled after fiasco in Courland instead of being backed up by a military force as he demanded). So we have a happy couple living in a relative misery for a couple of years (Anne as a duchess was
really poor but perhaps the local nobility would agree to spare some cash for Maurice) and then Peter II is dead and Supreme Privy Council has to address a succession crisis. Let's also assume that
there are no children in that marriage and almost definitely Maurice is not playing any noticeable role in governing the Duchy just because these functions had been taken out of his wife's hands.
In OTL Anne was an attractive figure because she was not married and looked as one who is going to be easily manipulated but even if she is married other options do not look more attractive.
Peter's grandson (future Peter III) means continuation of the Holstein-oriented policy which, during the reign of Catherine I resulted in a dangerous cooling of the relations not just with Denmark but also with the Brits and Dutch (and in Sweden position of the pro-Russian party became rather shaky as well).
Catherine of
Mecklenburg-Schwerin lived in Russia with her daughter but never was officially divorced and there was a fear that influence of her husband could get Russia into one more trouble (in OTL Peter I backed him up with the troops against his subjects but was forced to evacuate the troops under international pressure).
Praskovya Ivanovna was married to a general prince Ivan Dmitriev-Mamonov which would be considered inappropriate (to be married to a subject was a "no-no" for a Russian tsarevna).
Elizabeth Petrovna seemingly did not generate at that point any noticeable enthusiasm.
So let's assume that Anne is chosen just because all other options are not better and the fact that Anne is childless speaks in her favor. So far Maurice is mostly known for his "non-governmental" interests and is not considered as being a clear and present danger. Anna is given "conditions" limiting her future powers but upon arrival to Russia finds out that "we the people" (aka nobility in general and especially the Guards) prefer absolutism. So she is an absolute monarch with her consort (who is not an emperor) just being one of the greatest military minds of his time.... In OTL the #1 military figure in Anne's Russia was Munnich - a talented engineer and a good military organizer and reformer but not a very good (if reasonably lucky) general. Even his seemingly successful campaigns against the Crimea and Ottomans ended up as strategic flops due to the huge unnecessary losses, logistical problems, etc. Now, with Maurice being in charge of the campaigns, Russia may end up with a much earlier annexation of the Crimea and perhaps occupation of Bessarabia. What could happen after Anne's death is quite interesting because, unlike Munnich, Maurice was a figure that could endear himself to the Guards. Well, how about him marrying Elizabeth?
