OK. I have recreated the publication and subscriptions several times throughout all of last week, but I did it again just to be sure...
However, this time I did make a change that I had not made before.
Under the article's properties (for a Table) on the "Merging Changes" tab, under "Multicolumn Updates"... I unchecked the default property "When merging, apply changes to multiple columns in the same row in one update statement."
I always kept it checked because I thought it was the most efficient way for SQL Server to do the updates. However, unchecking this option has allowed the replication across the companies to proceed without the above mentioned error regarding the timestamp field.
I still do not know why SQL Server can ignore the timestamp field when individually updating columns versus a single update statement, but this seems to have solved the problem.
Thanks for the help and the suggestions.