You’ll have seen that there was a plan to have something like this but that was a while back:
No news on this it seems. Meanwhile, what @luke1 said. Any shortcuts, post 'em here!
If you were really desparate I suppose you could use the data file as a base, then trawl upwards through company numbers. You might check them at the companies URI (http://data.companieshouse.gov.uk/doc/company/{company_number} ) as that’s even simpler than the API and possibly not (so) rate-limited. A wrinkle is that if you wanted more than vanilla England and Wales companies - you’d need to do this for each of the various company prefixes. Most listed in the appendix at https://www.gov.uk/government/uploads/system/uploads/attachment_data/file/426891/uniformResourceIdentifiersCustomerGuide.pdf - note it’s missing “RS” and “BR” (uk establishment of foreign company so a meta-number?) and maybe others. Only a few types are likely to get frequent updates.
I don’t know if numbers are always sequential but apparently mostly should exist - see for example: