First, let's notice that there's no IN (fee فِي) at the start of 2:197, so it cannot mean that el-Hajj is IN anything. The proof for this is 22:28 where IN was used with eyām ma'loomāt فِي أَيَّامٍ مَعْلُومَاتٍ. This means that el-Hajj IS ešhur ma'loomāt أَشْهُرٌ مَعْلُومَاتٌ and we have to figure out what this means.
Passive ma'loomāt is a feminine plural and it's an adjective of ešhur here. The passive participle means that some verb was done on its subject if there's no active participle, or it means that some verb is doing on its subject in case that there's an active participle. Since we can't see an active participle we should conclude that ma'lumāt already have their passive state. We should note that the language of the Qur'an has an anticausative also, which is expressed by verbal form VII. So, who made ešhur ma'lumāt?
We should note that ešhur is more likely masculine, especially in 65:4 since it has feminine number (3) or 9:5 where probably masculine adjective Hurum is used. Due to this we could conclude that ešhur, as all other words, has multiple meaning and that some meanings are masculine and some feminine, because it's obvious that in 2:197 ešhur should be feminine plural. As ešhur in 65:4 most probably means "(3) months" where it's masculine, we can conclude that ešhur in 2:197 has some other meaning.
Root šeen-ha-ra describes revealing of something step-by-step, or in case of the Moon by phases. Ešhur as a collective noun can mean all these steps or phases, thus in relation to the Moon it means "(lunar) month". This collective form is singular and plural same time, as, for example, enfus is singular and plural for "self" (selves). This also mean that ešhur is not a plural of šehr, which has its normal plural in šuhoor, as, for example, beyt has buyoot.
If ešhur are not months, then what it means here?
Due to its root meaning and noun's form we could conclude that ešhur is some sort of a collection of revealings. Since el-Hajj is debating, discussing we could conclude that ešhur means a collection of presentations speakers/participants in el-Hajj. Why they're ma'loomāt? Because the host (Ibrahim and people after him) already chosen and decided which set of topics and their presenters will be el-Hajj.
The rest of 2:197 talks about those who participiate IN THEM (fee hinne فِيهِنَّ), meaning speakers/participants in presentations of the Hajj.
I hope this makes sense...