@jdaviescoates A platform to consider is SendFox. They offer a pay-once option for $US49 that would handle 5,000 subscribers. I can't vouch for deliverability, but I have referred several clients to this platform without any negative feedback. Another option is Mailerlite. Again, I have no firsthand knowledge about deliverability but clients are using this with no complaints.
That being said, ListMonk looks like an interesting platform that we are evaluating to provide email list subscription services (using Cloudron) to our clients rather than referring them to these external solutions.
Since your client is paying in Canadian dollars, I would suggest that they confirm with any of these solution providers that their platforms comply with any Canadian spam and privacy regulations.
Lastly, it can be difficult to migrate from one provider to another. The new provider can get prickly about "opt-in" requirements and whether a CSV list exported from Constant Contact (assuming this is possible) meets those requirements. Better to know what is and what is not permissible before leaving Constant Contact.