If they all download said book and then the book gets leaked and shared bypassing the library by your students in such a way that it ends up on multiple free websites while having electronic stamps of your library, it will be pretty bad. The publishers might view it as a malpractice to put it mildly. The best option is to consult with your colleagues, the library manager, and possibly, with people from the Springer portal you intend to use so heavily. I would not be too concerned since the students have access to the library and are actually encouraged to use its material, including said portal. If students somehow violate rules or share/spread copyrighted material without permission, they are solely responsible unless you yourself participated in such a malpractice. The probability of such a bad outcome approaches zero here. Many students would actually want to buy the hard copy version of said book. And the Springer publishers should only benefit. So I would not be too worried but you have to talk to your colleagues and to the library manager to make sure everything is “in line”.
Your idea of solving the problem for students is commendable. I would further say that giving students a couple of options (recommending at least two good books that are in line with your course) will also be a good idea. They will benefit from different approaches, especially if one book is more advanced/comprehensive and the other book is slightly shorter and easier. For main two semester courses it would be good if most students had hard copies of the textbook. For graduate courses, it’s good to recommend more books and expect students to do more research-like work.
Apart from all that, I would not neglect other publishers (e.g. Pearson) who create a lot of great textbooks in various fields. Yes, many of such books are expensive and new editions often appear on the market at exorbitant prices. However, it’s not necessary to always stick with the latest (expensive) edition making it tough for students to buy a hard copy. The other extreme is to completely discard all more of less pricey options i.e. to ignore even previous editions of expensive texts at discounted prices. I mean if we don’t buy and don’t recommend such books (most of them are very good), how are we to create them and improve on them? Please note, I’m not saying that Springer books on math are not excellent. They are.
Bottom line: Are there any rules which prevent me from assigning this as the textbook for the class and giving them the option to download it for free through the library? – No, unless you give your own access to the students or spread your personal copy among them or your students do something to that effect.