As @dotmiracle has already shown, there's some confusion as to which connector you're referring to; my assumption was that you're talking about the FPC connector on the logic board.
However, dotmiracle does bring up a good point; you have tried a different battery, right? If not, you definitely should, since it is possible the bad connection is on the battery connector rather than the motherboard plug.
Otherwise you're absolutely correct that the next logical step is to replace the FPC connector on the motherboard. If you've got someone with a hot air station you could try to reflow the connector; i.e., you melt the solder thus restoring any connections that have been interrupted due to a cracked solder joint. It's only slightly less work than actually replacing the connector and there's no guarantee it'll work, but it's something you could try.
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Have you tried cutting out something thin to sit on top of the battery connector so as to put more pressure on it when the cover plate is screwed down onto it? That could be a temporary fix while you order parts and/or locate a repair facility.
Hi Sem,
As @dotmiracle has already shown, there's some confusion as to which connector you're referring to; my assumption was that you're talking about the FPC connector on the logic board.
However, dotmiracle does bring up a good point; you have tried a different battery, right? If not, you definitely should, since it is possible the bad connection is on the battery connector rather than the motherboard plug.
Otherwise you're absolutely correct that the next logical step is to replace the FPC connector on the motherboard. If you've got someone with a hot air station you could try to reflow the connector; i.e., you melt the solder thus restoring any connections that have been interrupted due to a cracked solder joint. It's only slightly less work than actually replacing the connector and there's no guarantee it'll work, but it's something you could try.