No, but there's an alternative: using (Drawmij's) Instant Summons
The spell uses phrasing such as "the miniature replica", which already strongly suggests that there is a 1:1 relationship between miniature and secret chests. More importantly, the following passage makes it implicitly very clear that there is exactly one chest associated with each casting:
This effect ends if [...] the smaller replica chest is destroyed [...]
Beyond that, spells only do what they say they do. Therefore, since nothing suggests that a backup of the replica chest can be made, it can't, if you're following RAW (rules as written).
That being said, your DM can of course make a house rule that you can create a backup. Based on my experience, I doubt this would be a problem in terms of balance.
However, there is another solution that is possible while following the official rules.
You can combine your casting of (Leomund's) Secret Chest with (Drawmij's) Instant Summons.
Unfortunately, this requires a higher character level, as Instant Summons is a 6th-level spell, as opposed to the 4th-level Secret Chest. It also requires an additional 1000GP that are lost if you ever do lose your replica and have to (attempt to) recall it.
If both spells are within your capabilities, however, you can use Instant Summons on the replica chest, as it falls within the limitations of the spell:
You touch an object weighing 10 pounds or less whose longest dimension is 6 feet or less
It's not entirely clear what the dimensions and weight of the replica are, but given that even the big chest (3x2x2 feet) fits the size criteria, we would only have to worry about the weight, and I personally don't imagine a "tiny replica" to weigh over 10 pounds.
Fortunately, we can also calculate it (tl;dr: when made out of realistic materials and not solid gold, the chest likely does fall within the weight limitations)
We know the big chest is worth 5000gp, and the tiny chest 50gp (=1% of the bigger value). Assuming linear scaling, we know that the replica chest is 100 times smaller, though thanks to the square-cube law, we can't just divide each length by 100. Instead, we divide each length by the third root of 100, which is roughly 4.6.
This leaves us with a 0.65x0.43x0.43 feet chest (volume 0.12ft³, which is 1% of the bigger chest's volume, validating our calculation). Frankly, that's actually bigger than I would have imagined it.
Going forward, calculations will be made using a size of 20x13x13 centimeters because Metric is superior :D
Let's assume the chest is completely rectangular and has a wall thickness of 1cm. This means we have 2x(13cmx13cmx1cm) and 4x(18cmx12cmx1cm) sheets of gold to represent the chest walls, or 338cm³+864cm³=1202cm³ (as opposed to 3380cm³ if the chest were solid throughout.
By calculating volume * density, we get the mass, and by calculating using the density of gold (19.3 g/cm³), we know the chest is at most 23 kilograms. That's more than permitted, but the chest is for sure not made of 23 kilograms of gold, as that would make it significantly more expensive than 50GP. Assuming the chest is made of several materials including mostly oak (~0.9g/cm³) as well as some ivory (~1.9g/cm³) and gold (19.3g/cm³), let's average this out to about 2g/cm³, resulting in 2.4kg or less than 10 pounds. With 10 pounds = 4.5kg, the maximum allowed density would be about 3.7g/cm³.
Once cast on the replica, you can then simply teleport the replica to yourself by crushing the associated sapphire (within the limitations as listed in Instant Summons' description).
If you need more backups, you can just create more sapphires associated with the replica, as nothing states that only one sapphire can be linked to an object. This can be especially beneficial not only if you want to store backups in multiple locations (obligatory tangential advice to adhere to the 3-2-1 rule), but also to ensure that in the case of theft, you don't risk crushing your only sapphire while someone is actively carrying the chest. Instead, you can subsequently try again, perhaps using scrying for better timing, as you now know from the first attempt who stole from you.