The Sword Coast Adventurer's Guide is, in fact, already allowed material for Adventurer's League. It does not appear in the current AL Guide because it was published after the current AL Guide. However, AL put out an announcement about it. You should read the link, but:

Everyone, regardless of story origin (season) has the possibility of using the options presented in the Sword Coast Adventurer’s Guide. Add the allowed portions of the Sword Coast Adventurer’s Guide to the list of allowed rules for every story origin (see below).

Also, note that:

The D&D Adventurers League Player’s Guide will be updated with this information next season.

In other words, they're not going to bother with a revised Player's Guide this season, but from next season onwards the Player's Guide will include SCAG as an allowed option.

There are no rules on how long it takes to read a book exactly

Nowhere in the rules does it give any kind of guideline for exactly how long it takes to read a book, so that means that it would have to be something that you as the DM would have to decide.

However, I think there are rules/mechanics that are even better for what you are asking than a rule for book-reading speed.

Instead, use the Research and Training Downtime rules

So, you ask about how long it takes to read a book, but it seems the true core issue you are having is if there are any mechanics that can provide a balanced way for a character to use books to gain knowledge/proficiencies.

The good news is that there is something in the rules already that seems pretty close to what you might want.

Both the options talked about are listed in Xanathar's Guide to Everything Chapter 2: Downtime Revisited.

For researching the weapon, use the Research option

The research downtime activity allows a character to delve into lore concerning a monster, a location, a magic item, or some other particular topic.

This seems to thematically fit the exact problem you want solving.

The rules then go on to describe the time and cost associated with doing such research:

Typically, a character needs access to a library or a sage to conduct research. Assuming such access is available, conducting research requires one workweek of effort and at least 50 gp spent on materials, bribes, gifts, and other expenses.

And then they describe how that would resolve:

The character declares the focus of the research — a specific person, place, or thing. After one workweek, the character makes an Intelligence check with a +1 bonus per 100 gp spent beyond the initial 100 gp, to a maximum of +6. In addition, a character who has access to a particularly well-stocked library or knowledgeable sages gains advantage on this check. Determine how much lore a character learns using the Research Outcomes table.

As DM you can of course modify these rules as you see fit (for example you can waive or modify the cost if not appropriate for your situation). But the time it takes is a very useful abstraction of how long it takes to read and learn from books and resources.

Depending on how well the character rolls on their weekly check, they can learn up to "three pieces of lore" per week. Now, you may want to play around with this a bit but you can translate "pieces of lore" into a more precise mechanic for your situation pretty easily.

For example, you might break down the research into X number of questions (pieces of lore) they have to discover the answer to in order to construct the weapon. You can keep this number abstract or you can even define each question if you want to add some flavor (e.g. What materials do I need to make the weapon?, What processes do I need to construct the weapon?, etc.).

For gaining proficiency with the weapon, you can modify the Training option

Given enough free time and the services of an instructor, a character can learn a language or pick up proficiency with a tool.

The mechanics say:

Receiving training in a language or tool typically takes at least ten workweeks, but this time is reduced by a number of workweeks equal to the character’s Intelligence modifier (an Intelligence penalty doesn’t increase the time needed). Training costs 25 gp per workweek.

Now a weapon is not a tool or a language so you are going to have to allow this option to also cover this, but for your purposes I think it can work well.

I wouldn't recommend allowing this option to be used to attain weapon proficiency outside of this one plot-necessary scenario, so if you think that might be an issue with your players trying to use the mechanic later for other weapons, maybe make it clear that this is a special rules application.

Like the Research option you can modify these rules to tweak for your exact situation. For example, note that I don't think you have an instructor in your situation so you may want to modify the rules to account for this. The rules even go on to give a table of complications which would need to be modified to be thematically appropriate to weapons training, but would probably add some flare to the task if you wanted to add them.