If understand you correctly you need to drill under your path?
If this is the case I can’t see how you would drill a horizontal hole under your path without first digging a pit (at both ends) to give you the vertical planes you’d need to start off the hole and finish it.
As for drilling the hole…
I would think that a specialist drilling company, something like a diamond drilling outfit, would be able to help but at a hefty price.
Otherwise, and this supposes that you do have a way to access a vertical face to drill into below the level of your path, you would have to jerry rig a boring bit.
You could buy a diamond tipped bit similar to a forstner bit and have the shank welded onto a 20mm steel bar and file/grind four flats on the end to allow it to be held in the chuck of your drill…
(A small steel fabrication company would be likely to help you with this…there’s a decent one just off the 406 at New Southgate behind the new garage)
At this point I should say that I doubt a “domestic” type hammer drill would have enough power to achieve a five foot bore, and you’d probably need a decent SDS type tool…but HSS have these or you can buy a quite cheap one from
SCREWFIX.co.uk
Over such a length your problems are going to the effective removal of the considerable waste and heat generation (especially when cutting the brick and concrete)…
As it’s a hole, and the kit would be jerry-rigged, you couldn’t really run a water cooling pipe along the bore as the there wouldn’t be space to allow it…so it would have to be a very slow cut rate with frequent “pullouts” to clear the waste.
Is it not possible to lift only those tiles along the line you want, dig a narrow trench (say 40mm wide), lay the pipe, backfill, tamp, and then re-mortar the tiles in place over the top?