Use relative XP
As was indicated in the question, (MM, 84):
When a dragon becomes a shadow dragon, it retains its statistics except as described below.
Challenge Rating is not listed as one of the exceptions.
A strict reading would indicate that you keep the same CR when the dragon transforms. The Monster Manual describes what a statistic is on page 6:
A monster's statistics, sometimes referred to as its stat block, provide the essential information that you need to run the monster.
Under the broad heading of Statistics you will find many different stats including Challenge Rating (MM 9, under sub-heading Challenge).
So the rules indicate the CR stays the same, however the authors have provided a template which increases the CR of the Young Red Dragon from 10 to 13 after it was transformed to a Young Red Shadow Dragon.
Why the discrepancy?
I don't know. But it's there. More importantly, if it's an oversight, where did the authors make the error? Is the error in not including CR as an exception, or is the error in changing the CR in the template? Or maybe there is no error and I'm missing something (?).
Considering that CR tells you how great a threat the monster is, (MM 9) we need to determine if the stats gained by the Young Red Shadow Dragon make it more formidable than the standard Young Red Dragon. Once we determine this we can know where the error is in the text.
What makes the Shadow Dragon more formidable than it's former self?:
- If the Shadow Dragon is in dim light or darkness it has resistance to all damage types except force, radiant or psychic. That's a huge benefit.
- With its hide abiity as a bonus action in dim light or darkness, and extra stealth abilities (+8) it is far more likely the Shadow Dragon will surprise a party of adventurers.
- The Shadow Dragon's breath weapon instantly kills a combatant who falls to 0 HP, removing the death saving throw mechanic.
- And an Undead Shadow (MM 269) immediately rises from that dead combatant's corpse to do the dragon's bidding.
The only negative impact is that the Shadow Dragon is at disadvantage on attack rolls if it is in sunlight.
The Young Red Shadow Dragon is more formidable than the Young Red Dragon.
It seems that the error is in not including the CR in the list of exceptions. Is the Young Red Shadow Dragon deserving of +3 CR? Yes, this seems about right, however as a general rule I would not add +3 CR to a Shadow Dragon. A more accurate adjustment would be to take the relative XP increase from the template and apply the CR based on the XP increase.
Doing the math.
The standard Young Red Dragon has an XP of 5,900, while the Young Red Shadow Dragon is indicated as having an XP of 10,000. The XP has increased by a factor of 1.694915, that's 10,000/5,900. I'd call it 1.7. Apply that to the 450 XP of the Black Dragon Wyrmling and you get 450 x 1.7 = 765. Looking at the XP/CR chart at the front of the MM shows us that 765 XP is a CR of 3.
The Black Shadow Dragon Wyrmling would have a CR of 3.
I can guarantee you though: that beast is staying in its lair and never going out in sunlight without SPF I don't think so.
One final point is that according to its description, The Shadow dragon might retain or lose any or all of its lair actions or inherit new ones, as the DM sees fit. This could also increase or decrease the CR and would depend on the changes made. Of course this only applies to Dragons that already have a lair action (or does it? But I'll leave that for another question).