There appear to be a few services listed on this website:
https://directoriodecarga.com/bolivi...s-y-depositos/
If you find a suitable storage service, kindly share your experience.
A bonded warehouse is called "Almacen Fiscal".
The service is typically found near seaports. Given Bolivia is a landlocked country, you can forget about looking for one there. Fun fact, Bolivia has a navy...think Lake Titicaca. 
The other place you might find a bonded warehouse is near an airport. If there isn't one near La Paz, then I'm guessing there won't be one in any of the smaller airports...
I haven't heard of a storage service at the Bolivian borders, but I know the service can sometimes be found in the no man's land between countries or very near. The concept is, that you are not storing a bike in any country per se, but between countries. These services do not require a TVIP at all, and the storage hypothetically has no end date. There is such a service near the Caribbean between Colombia and Venezuela. I met a guy who stored a bike there. There were two options, storage under cover and in the open air. The former cost a little more. The "classic" storage charge is $1.00 per day (YMMV) and does not suspend liability insurance.
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Peter B
2008/09 - NJ to Costa Rica and back to NJ
2012/13 - NJ to Northern Argentina, Jamaica, Cuba and back to NJ
2023 - Peru, Brazil, Paraguay, Argentina, Chile, Bolivia...back to Peru.
Blogs: Peter's Ride
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