I did the same timeline check, and concluded that Egwene's TAR visit was ~2 days before Rand stayed with the Tinkers, just as you did. And the bottom line is, I don't think "Need" in TAR can predict the future (i.e. know that Rand would be at the Tinker's camp 2 days hence, and therefore steer Egwene there 2 days early).

However, I do think the Tinker camp Egwene visited was one of the many outside Ebou Dar, just like the one Rand stayed in (not necessarily the same one though). Why? Because Egwene commented on being in a stand of White Aspen trees. And, Wikipedia says White Aspen/Poplar trees are native to Spain, Morocco, and other moist, hot summer climate regions in Europe, Asia, and N.Africa. Which would fit with Ebou Dar.

Plus, we learned in KoD that Tinkers are gathering in / migrating to Seanchan-held lands in the south (and especially around Ebou Dar) due to the safety Seanchan law provides them.

I realize it's pretty flimsy "evidence" but there's not much else to go on.

This makes me think that maybe Egwene's future contact with the Seanchan (we know a Seanchan woman will save her, from one of her Dreams) will include Tinkers as well (maybe as peace brokers?), and THAT is why Need took her there (and why it was mentioned in TGS). At the time she was using Need, she was thinking about how to reconcile the WT, make it whole and strong again, etc., and I can see how needing to make some sort of resolution with the Seanchan plays into that.

Total speculation, I know, but I can't think of anything else. I doubt it has anything to do with any role the Tinkers might play in the Last Battle. Rand might "Need" to discover that, but I don't really see why Egwene would, especially when you consider what she was thinking about at the time she used Need - i.e. the White Tower, not the Last Battle. Whatever she needs of the Tinkers, it has something to do with the WT specifically, I think.