I'll start by noting that I only bought the [item name] because my
first pick — [first choice name] — doesn't ship to [country].
This is the more natural and correct way of saying this because the emphasis is on the only. Your reader will understand quickly that you only bought the [product name] and will be expecting the second clause 'because of [reason]' to explain why you bought that item rather than doing something else.
I have replaced [item] with the noun elephant in the next part for the sake of article clarity. The following sentences are saying that the company that sells elephants will not ship to a particular country (France):
The company doesn't ship elephants to France.
The company won't ship an elephant to France.
The company doesn't ship to France.
If you were to say:
Because the elephant doesn't ship to France
Depending on context, people would understand what you meant. However, more strictly, you could be saying that the elephant is the one doing the posting of items (similar to 'the company' in the previous sentences) rather than being contained in the shipment.
Because I can't get an elephant shipped to France.
Because I can't get [company name] to ship an elephant to France.
Would be a more correct way of saying you can't ship a particular item to a place.
A possible alternative to your sentence could be:
First, I'd like to ['highlight' or 'point out'] that I only bought the [item name] from [second company name] because [first company name], my prefered retailer, doesn't ship to [country name or 'my country'].
But it depends on the context of the sentence / the formality of the situation.