Our ZCS refuses to accept mail from MX backup service

[sunpower]

Hello everyone,

I am running Zimbra for a small non-profit organisation. For some reason our mail server misbehaves a couple of times a week (the reason of which is still unsolved). Usually this happens when I am away somewhere and do not have a computer in front of me to correct it. Therefore I orderred a MX backup service,which would hold the emails until the system is up again.

The problem is that zimbra just won't except the mail coming in from the MX backup. When testing the setup I get an error "The error that the other server returned was: 554 554 5.7.1 <xxx@ourdomain.org>: Relay access denied (state 14). "

To minimize spam I have previously ticked the following settings in ZCS Global MTA options:
enable DNS lookups, reject_invalid_hostname, reject_non_fqdn_hostname, reject_non_fqdn_sender, reject_unknown_client, reject_unknown_hostname, reject_unknown_sender_domain, and use six RBLs (sbl.spamhaus.org, dnsbl.njabl.org, cbl.abuseat.org, dnsbl.sorbs.net, relays.mail-abuse.org, bl.spamcop.net). So practically all spam prevention measures are in place.

I have tried to search the forums for guidance on this setup, but have not found anything which I could understand to relate to this.

We are running release 6.0.12_GA_2883.UBUNTU10_64 UBUNTU10_64 FOSS edition.

Does anyone have any ideas what I should change and where, please?