The DM probably doesn't want you to know yet…
It sounds as if the DM would like the party's current plane be a mystery that's solved over the course of the party's adventures there. If this is the case—and, given the constraints the party's working under, it certainly sounds like it is—, it won't matter the steps the party takes to determine the plane the party's on because the party can't learn that information until the time is ripe. In short, DM fiat prevents—and will continue preventing—the party from gaining this information until the DM says the party's (l)earned it.
…But if you insist
While actual full-blown planar travel—à la the 5th-level Clr spell plane shift [conj] (PH 262) et al.—is restricted, rare is the DM that also restricts extradimensional spaces, yet each extradimensional space is its own demiplane. With this in mind, have the party enter one of its extradimensional spaces—I recommend an enveloping pit (Magic Item Compendium 159) (3,600 gp; 0 lbs.) or something similarly (ahem) spacious—and, while therein, cast upon a party member the 4th-level ranger spell implacable pursuer [div] (Spell Compendium 120–1), having the party member first lower his spell resistance (if any) then voluntarily forgo his saving throw. The spell's effect enables the caster for 1/hour per level to know the subject's
location as long as the subject is moving, no matter where it goes on the same plane. Even if it leaves the plane, implacable pursuer tells you what plane the subject creature went to. (121)
Then have that party member exit the extradimensional space. The caster of the spell implacable pursuer should then learn immediately what mysterious plane the party member has ventured to.
Note that the party'll probably tip off the DM to this plan as the party'll have to acquire a scroll of implacable pursuer since I doubt that the party includes a level 14 or 15 ranger (or a level 7 or 8 assassin or consecrated harrier but I guess there could be a level 8 mystic ranger). And, once the DM is tipped off, I suspect that this otherwise surefire method, too, will fail capriciously.
Much more expensive will be a scroll of the 8th-level Sor/Wiz spell discern location [div] (PH 222) and casting the spell on a party member. The spell's effect
reveals the name of the creature['s] location (place, name, business name, building name, or the like), community, county (or similar political division), country, continent, and the plane of existence where the target lies. (ibid.)
Thus, unless "direct intervention of a deity keeps [the caster] from learning the exact location of" the subject—an unlikely possibility given the restrictions on spells dealing with deities—, an answer should be forthcoming. However, the DM may nonetheless lawyer even this, saying that when the spell reveals "the plane of existence where the target lies" the spell informs the caster that the plane is This plane right here… the one you're on now. If given such an answer, try to avoid doing anything rash.