Allowing a domain to send emails via another domain

Solution 1:

Assuming that both website1.com and website2.com have SPF records, add a:www.website3.com to each of those SPF records.

Edit: you write "So if I have 10 servers, it means editing the DNS zones for those 10?".

Yes, it does. Each domain's SPF record is authoritative for itself. However, if you have many domains that should all have identical records, you can change the first one, say website1.com, to include all the relevant information, then the others can reference it by defining each of their records to say just v=spf1 redirect=website1.com.

You'll still have to edit all 10 now, but you can avoid having to do it again later.