I agree, the circumstances of the Steeles invitation to Barton appear contrived.
My impression is the family at Barton hadn't even heard of them before.

After Edward left, Lucy would feel anxious to meet the Dashwoods-- and she did single out Elinor.

If Exeter is only 4 miles from Barton, quite likely the gregarious Sir John and Mrs Jennings often visited there.

Lucy may've stacked up a series of engagements to stay in Exeter till she met Mrs Jennings and Sir John, and contrived some relationship.
Once Lucy had the invite they didn't need to stay, so any real engagements with their Exeter host immediately gave way to leaving for Barton Park.

I can imagine the kindly Mrs Jennings, anxious to matchmake for others, feeling an obligation to help young, less well-off relations; even as she soon recognizes Lucy for a 'sly little thing.'

If Lucy and Anne visited friends or a relation in Exeter before, it wouldn't be beyond Lucy's capacity to plan for obstacles like money and chaperone. The Steeles likely jouneyed on stagecoach.
As a suggestion, their Exeter host may've sent a servent to Longstaple to meet and chaperone them to Exeter.

The calculated meeting really is an interesting idea I'd never thought of before.