Messages from certain senders vanishing - change mx priority to investigate?

[#1]

A client has a problem in that messages from certain of his customers fail to arrive. The client presently collects mail via POP3.  It's hard to tell what's happening here as there is no trace of the messages at all.
I want to (temporarily or permanently) change the system over to use direct SMTP delivery to an internal mailserver, but with a POP3 fallback via Plusnet.  Main reason is that this should give me the chance to find out, from the server's logs, what the problem is. (Chances are the problem is with the sending server being misconfigured or blacklisted, but of course try telling them that!)
However, I find that if I add a higher-priority  MX record for the internal server than the  mx.avasin.plus.net record it has no effect, and a domain lookup still shows only the one mx.avasin.plus.net record. Any suggestions how to make this work?
I did consider trying changing the email delivery to 'I will specify mx records' - but will this disable the POP3 service altogether, or will it still work as a fallback if I put the mx.avasin.plus.net record in manually?

[#2]

Before you mess about with mail settings, have you had one of the senders cc the email to another mailbox (e.g. gmail) so that you can verify that they are sending to the correct address?
Step 2 would be to totally turn off spam checking in case the sender has a bad reputation.

[#3]

Spam filtering has been turned off.
I want to switch them to direct SMTP anyway, but retaining a POP3 fallback would be a good idea, since an SMTP fallback is of no help if the broadband line fails. With a POP3 fallback the retained mail can be collected from any IP. If this presently isn't possible I may put it forward as a suggestion.

[#4]

That used to be possible, but was removed from the offering back in 2011. It's still possible to get SMTP mail delivery, just without the POP3 backup.

[#5]

Quote from: Anteaus

since an SMTP fallback is of no help if the broadband line fails.

Not strictly true.  Backup MX'es are designed to "store and forward" in the event that the primary MX is down.  However, you are correct in that this mail is actually unavailable until the Primary MX returns to service (in most configurations, anyway).
Personally, I would look at using a proper Backup MX service, rather than trying to rely on heath-robinson-ing the Plusnet system.  Dyn do a Backup MX service which is supposed to be fairly reasonable although I've not personally used it.
B.