You're welcome, glad the information is helpful to you. I strugged very hard in making my own book so I understand how painful the process is and how ignorant I am.
Almost if not all companies that have Indigo presses are commercial printing companies. At this point I feel the Indigo is one's best bet to print photo books, primarily because of the massive 53x75cm sheet size (beware: printable area is 51x74cm) the largest presses (10000 & 12000) are capable of, and also because they can accept Light Cyan and Light Magenta inks for printing. This otherwise inconsequential detail is perhaps the most vital contribution to getting superb photographic quality on the Indigo running the standard imaging engine and not the brilliant new HDLA upgrade (doubles the marking engine's resolution) that almost no one has. Combined, it would be one of the finest digital presses for high end book printing.
You must first accept that this means the print company is not prioritising interest in making a thing of great beauty that lasts a really long time, despite whatever they claim they do. I am not saying they defenestrate quality, but beyond spending a certain amount of time per project, it's no longer good business for them. As a client, you will have to accept some compromises. This was hard for me coming from a background of making the best quality original prints on fine paper with pigment inkjet printers I owned and controlled.
Try talking to them about the mechanical properties, archival qualities (ask, what pH is your glue when wet and what is the pH when cured) and provenance of their glue used for binding and you'll see immediately how stratospheric your "minimum requirements" are. Then try entering into a discussion about paper and be disappointed. It's not fun. This is why I decided to research and source all my own materials and work with a company that allowed me to direct the entire process and make my own custom adjustments unique to my project.
There are a few artisinal book binders running their own small specialized studio who may use archival glue, but none use archival boards for hardcovers that I know of. If you are willing to pay, they should be happy to get such material for you. Then be prepared for the binding service to cost many times what the printing costs, and quickly realise that it isn't a worthwhile investment.
Archival also needs to be defined clearly. As mentioned earlier, lots of so-called archival board are actually not constructed with truly stable materials. You will often need to dig deep into what the thing you are using is made of.
A softcover book simply uses thicker paper to make the cover. So just choose paper that your research indicates has the archival properties you seek - though that is easier said than done. Any paper for printing would also have to be compatible with the ElectroInk if you use Indigo. Not a big problem though, since there are now thousands of certified papers. Actually, big problem, because there are so many you don't know where to start.
Making a softcover would also usually mean perfect binding is used unless you consider other forms of more exotic binding, assuming the print company even offers that. The glue used for this kind of binding is virtually always a hot melt adhesive, and thus far I remain unconvinced about it's archival properties. Some hot melt adhesives are better than others. I find the feel of a softcover to be too cheap, flimsy and commercial to have the haptic qualities I desire for my own book. You will also the lose the lay-flat of pages when you open the book. This may or may not be a big problem (it was to me) Do consider how it feels in the hand.
I have found that despite my proclivity for matte fine art papers for inkjet printing, a high quality
coated matte or satin smooth-textured paper is best for printing photo books on the Indigo. Other people tend to like the super expensive Mohawk Superfine Eggshell for printing art books on the Indigo. I've found it to be a beautiful paper but like all uncoated paper, it develops a disgusting sheen in areas of high inking that is unavoidable unless a matte varnish could be applied on top. The disconnect of the plasticity glossy ink sheen to the beautiful, tactile matte paper ruins it for me. The total gamut and dmax is also quite a lot less. Furthermore, any texture that catches the light causes a glare to be visible and further obscures the printed image. When the gamut is already so limited compared to inkjet, losing any more can be quite unbearable. Custom profiles with excellent gamut mapping are highly desirable to deal with the mapping of out of gamut colours. There is also a nice trick to getting incredible dmax on the Indigo, but not all companies might let you with their setup.
I wonder since you are doing a small number of books, can't you do the printing yourself or get done by a good inkjet photo printing shop who would use the papers and inks of your choice (some are here on this forum) and then get the book bound separately at a book-binding shop.
Printing it on your own pigment inkjet printer seems to have everything going for it at first. It was the first thing I considered. But there are significant downsides.
1. It would take me about 2 weeks worth of printing just to yield one book, not including the binding and sourcing for those materials. These machines are slow compared to commercial printing presses. If you want to do 5-20 books, it quickly becomes silly. You could always use a lower quality printing setting to decrease print times to just 20-25%, but why compromise on quality now if you demanded for the best going down this route?
2. Another issue is that papers coated for inkjet don't fold well, so cracking is to be expected for sewn binding. It can be avoided with perfect binding but no lay-flat of pages when open.
3. Inkjet prints are not very durable. You will get scuffing and setoff (ink rubbing off and staining the facing page, usually happens to the more heavily inked areas) on facing pages over time. You can spray the pages with varnish, but if you know what that entails, you'll know why I didn't even bother considering it.
4. Durability again, it is very tricky to feed a sheet that is already printed back into your printer and not scuff the printed side. After all this work, the tiniest flaw might be quite upsetting. Don't underestimate how challenging this is especially with large book pages.
5. Inkjet printers of the kind we use don't have the mechanical means to ensure precise registration between the front and rear sides of the printed sheet, and also lack the means to allow you to make the adjustments necessary to ensure good registration. If your book design requires good registration, you'll be in a world of pain struggling with this otherwise insignificant issue.
6. The monetary cost of printing like this and using a specialised binding service is quite high.