Whilst it doesn't answer your original question CW, you can calculate the actual distance to be hiked from the info from the site, at least you can from Doogal.
I had a play with the aforementioned Doogal site and created a simple hiking route. Then downloaded it as a CSV. It gave me Lat, Lon, Elevation and cumulative distance for each leg. (A few random checks makes it appear to me that this is the horizontal equivalent distance.)
I loaded this into Excel and added columns to calculate a) the elevation difference for each leg, b) the leg distance in metres and c) the actual distance including the extra due to the gradient (I call it Grade Distance). Then pasted these values down the rest of the rows.
The Excel spreadsheet formula for the Grade Distance is =SQRT((E3^2)+F3^2), which is Pythagorus' Theorem. Cell E3 is from the Elevation Difference column and F3 the Leg Distance column, both in metres.
I did try the Mapometer site but it wanted me to register if I tried to save anything so I went no further with that. But you may be able to do something similar there if you can download the route as a CSV.
You may find the following table of help when planning routes (E&OE!). It shows various stats for a range of 10m elevation changes over 125m horizontal equivalent (HE) on a 1:25K map (5 millimetres in this case.) The Grade Distance is called Slope Distance in this table, the distance you actually hike. Count up how many 10m contours your route crosses in each 5mm and look up how much pain will be caused!
I can generate other scales if necessary. (Nerdy? Me?)
Code:
Gradients for Height changes in Metres over 5mm on the map
(125M HE at scale 1:25000)
Height Gradient Gradient Slope Slope Increase Increase 100 metres
Change (1 in x) Percent Angle Distance Over HE Over HE paced as HE
10m 12.5 8% 5° 125m 0.4m 0% 100m
20m 6.3 16% 9° 127m 1.6m 1% 99m
30m 4.2 24% 13° 129m 3.5m 3% 97m
40m 3.1 32% 18° 131m 6.2m 5% 95m
50m 2.5 40% 22° 135m 9.6m 8% 93m
60m 2.1 48% 26° 139m 13.7m 11% 90m
70m 1.8 56% 29° 143m 18.3m 15% 87m
80m 1.6 64% 33° 148m 23.4m 19% 84m
90m 1.4 72% 36° 154m 29.0m 23% 81m
100m 1.3 80% 39° 160m 35.1m 28% 78m
110m 1.1 88% 41° 167m 41.5m 33% 75m
120m 1.0 96% 44° 173m 48.3m 39% 72m
130m 1.0 104% 46° 180m 55.3m 44% 69m
140m 0.9 112% 48° 188m 62.7m 50% 67m
150m 0.8 120% 50° 195m 70.3m 56% 64m
160m 0.8 128% 52° 203m 78.0m 62% 62m
170m 0.7 136% 54° 211m 86.0m 69% 59m
180m 0.7 144% 55° 219m 94.1m 75% 57m
190m 0.7 152% 57° 227m 102.4m 82% 55m
200m 0.6 160% 58° 236m 110.8m 89% 53m
For changes over 10mm the gradients are half as severe!