My experience is with the Netherlands ' setup only but I imagine the Belgian rules will be fairly close as they are signatory to the Schengen convention etc.etc.
In NL it is impossible for a foreigner to get a bike registered in his name ( source :RDW) unless he has a t least six months of established residency in NL.
For a tourist to get a temporary registration in his name it would require him to get the temporary EXPORT plates and regs. and he would have to take the bike out of the country .
The only way around this, and I have an e-mail from the RDW the Dutch organ in charge of licensing suggesting this , is the practice I have used with success on a number of trips to Europe.
Namely you need a contact person who lives in the country you want to buy the bike (Belgium) either the seller, a friend or a relative who already owns and insures a bike. You simply buy the bike and leave /change the ownership of the bike to the name of the friend/relative.This may entail a small fee for a compulsory inspection. Then for insurance you may be able to have yourself written into the friend/relative's policy as a second rider or you can buy a separate policy under your own name. You can in the former case work out the pro-rated monthly cost of your bike insurance with your friend/relative who then terminates the double rider cover when you are done with the bike.. This gets you a green card that will cover all of Europe and north Africa and more.Either way it will be less trouble and cheaper than going the route of export plate rego and its fees.
Do take photocopies of the friend/relative's driver license and your own and the insurance document and both of you sign each as proof that you are using the bike with permission.I do but have never run into any problems with police asking for such-simply by avoiding getting stopped by them.
Any infractions such as photoradar fines will be sent to the address of the legal owner i.e. your friend/ relative and they may bring him unwanted demerit points against his license, so do be extra cautious (good) and do repay all the fines.Try as you will there will probably be one or two such fines,just keep the sped down $$$$$ in cities and watch the limit signs.

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