You can look for shared access to ahrefs, it's not soo expensive. There is the same option for semrush, though they have discrepancies in the link profile and I prefer ahrefs. But ahrefs also has a free backlink checker that shows the number of links and DR, so you can estimate the approximate number of links. You can write a parser that you give a list of domains to as input, and it checks through the free ahrefs tool. I used this when I needed to check hundreds of domains, and there was the weakest tariff ahrefs, without batch analysis - you pay only for solving the captcha, it came out to about domain = cent