Climbing line (or rope[d] line or belay line or simply line) would be my suggestion here based on your first example and the terminology normally used in English (although your second example does then pose a bit of a problem), with Hakenreihe technically referring to the vertical line of protection/anchor/attachment points on the wall (and not the holds setting the route).

Here are some examples. You can easily find many more.

We have over 120 lines of indoor climbing, covering over 2,000 square meters of climbing surface and reaching heights of 12m. We try to ensure there are at least 3 routes per line on every line, so at capacity our maximum number of routes will exceed 360.
http://bouldersuk.com/boulders/indoor-climbing/

Set up two bottom ropes of different colours, not on adjacent climbing lines but one line further on; the climber is on a route on the middle line, but the bottom ropes are from the adjacent lines to this central climbing line.
https://issuu.com/pesdapress/docs/climbing_wall_leading_samp...

103 Roped lines with over 350 routes
https://www.whitespiderclimbing.com/the-centre/
Liam Halsey is at White Spider Climbing today setting on lines 14 and 15!
https://www.whitespiderclimbing.com/journal/all/our-setters/

What type of climbing wall is being built in the new centre?
The wall is 13 metres high, 141 sq metres with seven bolt-on holds per sq metre. There are nine rope protected climbing lanes with a minimum of 18 routes.
http://www.dmu.ac.uk/documents/dmu-students/ask-us/leisurece...

The 17 belay lines have colour coded routes and it is possible to complete several different routes from just the one belay line.
http://www.lancaster.ac.uk/sport/sports-centre/facilities/cl...

The wall space will host approximately 25 different roped lines, each with a range of routes following coloured holds
https://www.mountainfestival.co.uk/3wm-climbing.html

The climbing wall is divided into nine roped lines, consisting of three routes per line.
http://www.jagssportsclub.co.uk/climbing-wall/roped-climbing...

When top rope climbing, only use top ropes for the vertical line they are set up on. Do not use them to climb adjacent lines.
Quickdraws are in place on all lead climbing lines. Do not tamper with these. On some climbing lines seconding or top roping is not permitted. These lines have been clearly labelled.
http://bouldersuk.com/boulders/termsandconditions/

The 12m high Climbing Wall comprises of eighteen climbing lines, with three routes on each line. Thirteen lines are real form textured surfaces and five lines are on textured panels.
Qiuckdraws are in-situ on many routes to allow lead climbing. Those which do not have quickdraws still have bolts to allow for quickdraw placement.
https://www.verticalgear.co.uk/directory/listing/sport-centr...

Other ways of expressing this in German include Sicherungslinien, Hakenlinien and of course Kletterlinien
Über 50 Sicherungslinien mit mehr als 100 Routen locken Klettersportler in Routen vom 3. bis 10. Schwierigkeitsgrad.
https://www.alpenverein-baden-baden.de/kletterzentrum/

100 Routen (49 Hakenlinien)
http://www.kletterzentrum-freising.de/index.php?das-haus

Indoor: 300 m² Kletterfläche, ca. 12 Meter Höhe und 17 Kletterlinien mit je 3 Routen pro Linie
http://salzburg.naturfreunde.at/service/kletterhallen-salzbu...

The best examples I have found illustrating the use of the term in German are on the following sites, with some very useful photos and explanations.

Mit Lila, der ersten Route in Hakenreihe 44, wartet auf Dich eine abwechslungsreiche Tour im unteren 7. Schwierigkeitsgrad.
Um keine weiten Wege zu gehen, Du bist ja schließlich zum Klettern gekommen, befindet sich in derselben Hakenreihe eine weitere Tour im 7. Grad.
https://www.magnesia-klettern.de/2017/12/fit-fuer-den-grad-7...
Die vierte und letzte Route trägt die Farbe pink und ist in der gleichen Hakenreihe verschraubt.
https://www.magnesia-klettern.de/2017/11/fit-fuer-den-grad-6...

The following is a link to a plan of the numbered climbing lines
https://www.magnesia-klettern.de/topos/

Der Kletterer wird im Falle eines Sturzes durch ein Seil gehalten, das an sogenannten Zwischensicherungspunkten (Sicherungspunkten/Haken) befestigt wird, die entsprechend in der Wand angebracht wurden
http://www.ontopklettern.de/ueber-uns/wissenswertes/seilsich...