The Knights and Ladies section includes extra lands you can be from. If you come from a land where polygamy is allowed, you can certainly do it.
The person who is likely to forbid you is your Lord. If your acts are public enough, it may come to the attention of your king.
Knights are sheriff, judge, and jury for all matters of Low Justice on
their own domains (see below for the three types of temporal
justice). If a character has land, then it is his responsibility to
maintain justice within it. Likewise, on their lord’s land, knights
must act on their lord’s behalf, either delivering justice then and
there or else taking the wrongdoer to the lord’s court. Knights who
break the law are subject to justice in the courts of whoever was
offended. If the law breaking takes place in a knight’s own domain,
then his own lord makes judgment. The only exception to this is when
a knight breaks a rule of his lord, in which case the other knights
of the court stand to deliver a judgment. Appeal to a higher court
can be made, but the higher lord’s court can always decide whether or
not to hear the petition. Knights accused of wrongdoing at any time
may claim trial by combat instead of normal court justice. (p22 core)
If you have the social pull and influence to get multiple people to marry you and your lord disagrees, you can also demand trial by combat to say you are innocent. If your lord (and eventually king) is fine with it, your word is law of the land, and you can decide whether polygamy should be legal. It's expected that you'll have lovers and concubines of course
Many children are born out of wedlock. Noblemen seem especially
subject to propagating this vice. Their partners are sometimes called
lovers, concubines, courtesans, or paramours, and are frequently of a
social class significantly lower than that of the nobleman. (p21 core)
If you are willing to forgo the manorhouse, it's a lot easier. As the book notes, taking concubines is common. So long as they're of a lower class they're not a threat to your political union, so it's more permissible.
And historically, polygyny was common, if detested by the church. The first wife may have some legal recourse if you try to take a second wife.
As the aforementioned paragraph 23 of the Cáin Lánamna suggests, the
cétmuinter was entitled to the coibche her husband gave to a new wife,
the adaltrach .Furthermore, a gloss to Heptad 6 allowed the cétmuinter
to inflict any non-fatal injury on her husband’s second wife for a
period of three days, while the adaltrach was only allowed to scratch,
pull hair, speak abusively or inflict minor injuries in retaliation.
You should definitely discuss these matters with your DM. Under the law, your first wife may well be able to 'accidentally' kill your second wife, and have you pay compensation.
p204 also has a table for any cuckolds. Your wife may decide to roll on that table and get vengeance on you, per the above historical notes.