I heard back from linkshare customer support and they pointed me to this article:
https://rakutenlinkshare.zendesk.com/hc/en-us/articles/201105906-Deep-Linking-Structure-Creating-Tracking-Links-Outside-the-Publisher-Dashboard
The Linkshare Automated Link Generator API appears to have been decommissioned.
However they have made it much easier to construct deeplinks now. It is much more like other affiliate networks where you no longer need to know the offerid and tempid. All you need is your Linkshare affiliate ID and the relevant merchant ID.
The new format deeplinks are as follow:
http://click.linksynergy.com/deeplink?id=<affiliate id>&mid=<merchant id>&murl=<urlencoded target url>&u1=<sub id>
Much easier than the old format which was like the following:
http://click.linksynergy.com/fs-bin/click?id=<affiliate id>&subid=&offerid=<offer id>&type=<deeplink type>&tmpid=<temp id>&u1=<sub id>&RD_PARM1=<urlencoded target url>
Both types of deeplinks still work but the second is very difficult to programatically create as the offer id and temp id are not that easy to work out and can change over time.