I do not see an example of a 10,5cm lFH Kp. 16 at the Bulgarian National Military Muesum link. I think you might be looking at a Krupp export 10,5cm. By the way, German designations for these howitzers is 10,5cm for those in Field Artillery service. The 10cm designation is only used when it is a gun (not howitzer) in Foot Artillery service.
I have no figures about export, but I think the gun may never have been exported as such. It was developed by Krupp at the request of the German forces and not as export weapon. During the war there were hardly any exports by Krupp, because it simply not allowed by the Imperial govt unless it served a political purpose. On the other hand, export guns that found themselves at the Krupp factory in 1914 were confiscated and when not in the right German calibres handed over to allies as military aid, not exported by Krupp itself. This happened for instance with the Krupp 75mm field guns for Brazil and several other models.
My guess is that all lFH 16 Kp were employed by the German forces, although they also ended up in Bulgarian service. That could have also happened after the war, during the Armisitice for instance, when the Imperial govt was handing out weapons to Allies and friendlies.
That Wiki page is well annotated and says that by the end of WW1 66 batteries were still in service. With four howitzers per battery that makes 264 pieces still surviving. Others may have been lost or indeed handed over to Allies like Bulgaria, but not exported in the legal terms. We would have seen them somewhere.
It was a good weapon, with a higher range , but heavier than the suiccessful Rheinmetall design. Could it have been heavier because the Krupp model used the carriage of the 12 cm L14?
And if someone knows how and when Bulgaria received them, that would help...
And like I suspected, the Bul. artillery inventory miraculously expanded between Sep 18 and Sep 19:
"However we can make some suppositions. In October 1915 Bulgarian Army had 1231 guns. In September 1918 it had 1395 guns in the field (British Official History : 1597 – 202 German guns). According with История на служба "Артилерийско въоръжение"…, p. 97, on 15 September 1919 the Bulgarian Army had 1956 guns. This means an increase of 725 artillery pieces. To them we must add the guns that were lost during the war because destroyed due to faulty ammunition, worn out and captured or destroyed by the enemy. In 1921 the Interallied Military Control Commission found in Bulgaria 1468 guns of different calibres and patterns (История на служба "Артилерийско въоръжение"…, p. 102 says 3400 guns, but it is certainly a misprint)."
The Treaty of Versailles in 1919 authorised a postwar German Army to have only 84 10.5cm howitzers and 204 7.7cm guns so there must have been a massive
number of artillery pieces to dispose of. The US acquired about 1000 10.5cm howitzers and put about 650 into reserve (Williford).
Indeed and these were AFAIK the Rheinmetall lFH16s that were deemed so good, that the US was inspired to start building their own 10,5 cm how., later known as the M101
The requirement for the US Ordnance dept to develop a light 105mm howitzer to partner with a light field gun, as the British and Germans had done, is one of the recommendations of the Westervelt Board Report which was produced at the end of 1918. I think putting the German 10.5cm lFH 16s into reserve was a quick way to satisfy this recommendation since development of a new howitzer could be predicted to be a protracted affair with the limited financial resources available for new artillery. To their credit the Ordnance Dept managed to produce a prototype 105mm howitzer, the M1920, and a number of experimental carriages which, over an extended period and many iterations, became the M2A1 (M101) in 1939.
Williard says in his book "American Breechloading Mobile Artillery 1875-1953" that the 600 or so lFH 16s in storage were very prone to rusting and although remediation programs were proposed nothing was done and the lFHs disappeared from official records after 1925, presumably scrapped. A number were offered as memorial guns to any community in the US that wanted one - this accounts for perhaps 50 or so. Presumably the remainder of the 1000 or so were scrapped. There were about 40 sent as war trophies to Australia from the UK - perhaps 10 have survived.