We are more than happy to assist, please keep us updated on the installation. I'm sure others in the community would have run into a similar situation and be interested to see your final installation.
Great advice @Iain, thanks for joining in the discussion.
This is the same solution we went with on a wall at the side of our house that is clad with blueboard.
Looking forward to seeing more of your posts soon and reading about your own projects and plans. Please don't hesitate to post anytime you have something to share or need a hand.
Attaching a board across the studs is most definitely the best approach. The forces on the clothes line are not just downwards, so merely "hanging" it on the wall would be a short lived solution. Good to see you have accepted Iain's advice. Emphasised, in case anyone else with a similar problem is thinking of shortcutting by just hanging something that also will be pulled at during normal use.
@TedBear might have a different preference from what I would suggest so it would be great to hear from them as well.
For fixing the timber to the studs I'd recommend Buildex 14-10 x 100mm Climacoat Bugle Head Batten Timber Screws - 25 Pack. For the clothesline use the thickest 40mm galvanised coach screws that will fit through the holes on your brackets.
Depending on how long the bracket is you might need to use two 90 x 35mm H3 MGP 10 Treated Pine Outdoor Timber Framing next to each other. @Jason looks like he has used a board similar to 140 x 19mm KD Select Grade Merbau Decking - Linear Metre.
Please let us know if you have additional questions.
For attaching timber into timber, I tend to go for screw length of 3 times the board thickness. In this case 3 x 35 would be 100mm screws. They would bite in ~ 65mm, which would be 2/3 of the likely upright you would be screwing into (i.e. probably 100mm). As far as attaching the line, I suggest 40mm length, as there isn't any point going much longer than the 35mm board thickness. (But that's without knowing the thickness of the clothes line unit at the place where the screw holes are. If it is thick metal or a tube, add 35mm to the thickness & round it to the nearest available size. I hope that makes sense.
Edit: I just saw Mitchell's reply, made about the same time as I was writing.... I am relieved to know we came up with same lengths, but he has kindly given you more details about the actual products to buy.
Thanks for your good-willed offer. I'm happy to know it helped... use the info for deciding future projects too. I'm having a drink of my choice (Guiness) while writing, so that will do. Thanks.