Lets consider the theory from first principles:
Neglecting losses, the inductor current builds up linearly storing energy in it for the on time which for a voltage doubling boost converter is ideally 66.7% of the period. It then cuts off and the inductor current decays linearly as it dumps that stored energy into the output capacitor. The peak current is therefore double the average current during the on time or off time and to deliver a continuous 1A out, with only 33.3% of the period to deliver the energy, means you've got to multiply by another factor of three to get the peak switch current then apply whatever fudge factor you need for losses and derating. You are therefore looking for a peak switch current capability in excess of 6A. (There are ways of fudging that, e.g. by not letting the current drop to zero before switching the inductor's swinging end to ground to dump more energy into it, but that reduces the effective energy storage of the inductor so what you gain on reduced peak current you typically loose on cost of magnetics and controller complexity.)
Also note that switching frequency directly impacts the size and thus cost of the capacitors and inductors required, so within reason, the higher the better. An integral synchronous rectifier is only a moderate benefit because its only worth up to 5% better efficiency at your output voltage. They are far more important at lower voltages and higher currents as diode Vf*If become the dominant losses.
Put in appropriate search criteria, sort by price, and you can work through from the low end viewing their datasheets and quickly rejecting those that obviously cant meet your requirements. I did a quick and dirty job on the limits and got about 60 candidates. It then took me no more than a minute to reject the first four based on the first page of their datasheet. Note whether they require an external diode for any you don't reject.
I reckon, ignoring price, you'll be able to get it down to around ten or so in no more than an hour. Then take the lowest and highest frequency ones remaining, and work out the cost of the required inductor and capacitor properly. Download the search results as a spreadsheet, flag the ones that *CAN* do the job, and in another column, if they need a diode, and interpolate a cost for the passives vs frequency from the two extremes so you can calculate a total cost for each of the validated candidates. Pick the cheapest total cost and work out its passives cost, and diode cost if any in a separate column, use the figures to refine the cost vs frequency interpolation for the others, rinse and repeat till you've got the three cheapest candidates.
IMHO there's about three hours work there to narrow it down. Further distributors with a decent parametric search with a download results option will be less work as you can use the reject list of parts that cant do the job you've accumulated so far to automatically eliminate them from the downloaded new results, and for anything in your qualified candidates list you only have to update the pricing and see if it shifts it into being a candidate for the cheapest three overall, so you only have to do the full costing exercise for any new candidates that cant immediately be ruled out on chip cost alone.
Dropping back to 'Clear's suggestion of an entirely discrete design - well, without custom magnetics and a fairly complex control circuit, efficiency will *SUCK*, and its a major design exercise to even get over 80%. Evaluating and optimizing even a single candidate design properly is likely to cost more than you can afford spread over so few units in the production run. (I estimate, ballpark, around $0.75 per design per unit over 3K units, with a subject expert on hand, far more if you are learning on the job and billing it to the project.) If you were planning to build several million or better several tens of millions it would be a different matter,
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