I want to know the reason for allowing kings to take multiple wives. Is it Rajo guna or any other reason based on Shruti, Smriti, Puranas?
According to the Smritis, how many wives a person can have depends on his Varna.
´There are four classes: Brahmin, Kshatriya, Vaishya and Shudra. Of these, according to the order of the classes, a Brahmin may have four wives, a Kshatriya three, a Vaishya two, and a Shudra one.
Baudhayana Dharma Sutras 1.16.10
Now, there is some difference of opinion regarding this.
According to the order of the classes, a Brahmin may take three wives, a Kshatriya two, and a Vaishya and a Shudra one each. According to some, one may even marry a Shudra woman just as one marries others but without the recitations of ritual formulas.
Vashishta DharmaSutras 1.23.
But in any case, a Kshatriya is always allowed to have more than one wife. And, a king comes mostly from the Kshatriya clan.
Now, if you think Raja Guna is the real reason, then why do you think a Brahmin (whom you must be associating with Sattva Guna) is allowed to have the highest number of wives among the four classes?
But you're very right with the fact that a king (as also the Kshatriya) is a manifestation of the Raja Guna.
Manu Smriti 12.46. Kings and Kshatriyas, the domestic priests of kings, and those who delight in the warfare of disputations (constitute) the middling (rank of the) states caused by Activity.
12.47. The Gandharvas, the Guhyakas, and the servants of the gods, likewise the Apsarases, (belong all to) the highest (rank of) conditions produced by Activity.
12.48. Hermits, ascetics, Brahmanas, the crowds of the Vaimanika deities, the lunar mansions, and the Daityas (form) the first (and lowest rank of the) existences caused by Goodness.
Here, "Goodness" refers to Sattva and "Activity" refers to Rajas or Raja Guna. So, the Brahmins come from the Sattva. But yet they are allowed to have more number of wives than the Kshatriyas are.
NO; the mode of Rajas has nothing to do with number of wives. It has obviously nothing to do with Kshatriya-s.
Rajas is associated with "desires". All entities have mode of Rajas in them in some amount. It's cause of various sins, as discussed in this answer.
BG 14.7 - Rajas is the nature of passion, which is produced from lust & attachment; That binds the embodied [soul] through attachment to action.
Besides, Kshatriya-s having multiple wives is justifiable because, they are responsible for the protection and maintenance of the kingdom. Kshatriya-s are more likely to get martyred during war time. Due to which the king & his associates tend to procreate more copies of them.
As per scriptures, for anyone there is no limitation in the number of wives. e.g. Krishna had 8 wives as discussed here.
A Kshatriya can choose wife from any classes, except Brahmana. For example, Dhritarashtra had 2 wives, Gandhari and another from the class Vaishya, who was the mother of Yuyutsu. Usually it's not recommended to choose a wife from the Shudra division for the other divisions, as discussed in this answer.
Certain Kshatriya-s like Rama had only 1 wife, while Bhishma didn't have any wife. But they can be considered as special case.