DIY sheds buy low quality stuff which they sell at high margin to people who are often relatively unknowledgeable, but they open late every night plus Saturdays and Sundays. Hence the (to me as a joiner) eye watering prices.
Timber merchants tend to have various qualities of timber in given sizes, for example in ex-3×2in you might find that they stock rough sawn whitewood framing softwood (knotty as hell), CLS (whitewood), PSE (or PAR - same thing really) whitewood, PSE joinery grade redwood, etc. Unlike B&Q the staff tend to be a lot more knowledgeable (providing you avoid the spotty herbert with his hands and eyes permanently glued to his phone). The open early, but close early, and whilst they may open on Saturday morning, forget about Saturday afternoon Sunday. They also observe bank holidays. In general their prices on singleton items are better than B&Q, and their quality is often a lot better. Also if you get into the £200 order area they may well deliver (possibly FoC) and they may start giving a bit of discount. Just avoid the guys who insist on you having a trade account.
The downside is that timber merchants often insist on you taking timber "as it falls", but the trade take account of this on bigger jobs by calculating in a 5 to 10% wastage factor. In carcassing the poorer stuff can often get used for noggins, solid strutting, temporary works and the like so less of a loss. The sort of stuff you want for this job needs to be straight, so give them a cutting list and tell them what it's for. Also, because you need straight timber sight along all the pieces looking for wind (twist), cupping and side to side bowing when you get them. You can always reject them and demand a replacement.
Some merchants don't cut, others do. I think you need one who offers a cutting service from the sound of it, so ask