If he was definitely a competitor, the only possibility would seem to be the Liège-Rome-Liège (Marathon de la Route), which started on the evening of August 16th and finished on the afternoon of the 20th. But at that stage the British authorities had not yet ordered UK citizens to leave Germany.
However, it may be a confusion/conflation with the International Six Days Trial, which was centred on Salzburg, and was scheduled between the 21st and 26th. By that time private motoring had been banned in Germany and petrol was only available to foreigners with special 'tourist coupons'. I do wonder if he's confused Luxembourg and Liechtenstein though, since most of the British returned via the crossing at Feldkirch.
http://speedtracktal...t-1939-germany/
By the evening of Wednesday August 23rd, the more than 60-strong British contingent were becoming increasingly nervous about their safety – the organisers were already talking about offering safe passage out of Germany – and one of the Army riders had received a telegram recalling him to his regiment. The tipping point came when it was learned that French citizens had been advised to leave Germany within 24 hours: even so, the British riders continued for the fourth day, while telegrams flew backwards and forwards between team managers, teams and embassies. German daily papers had stopped reporting the event after the first three days and early on the Friday morning, almost all the British competitors, support teams and spectators abandoned the trial and headed west through Innsbruck into Switzerland and France, many of them riding their trials bikes home in convoys of up to thirty: the Army teams and the four remaining civilian riders stayed a few more hours, but they too were on the road by Friday night, escorted to the Swiss border via Munich and Bregenz by a Wehrmacht colonel, apparently on the orders of Adolf Hühnlein himself. History records that the 1939 International Six Days Trial – contested for the Hühnlein Trophy! - was won by Germany, but British riders had held most of the leading positions until the time they abandoned the event, so it would have been a somewhat hollow victory and it also seems that the FICM never ratified the result.