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I wouldn’t do that in public personally speaking.
XEX West in Umeda. Super fancy and super pricey. Took my wife here for our anniversary last year. You can also make a reservation on line
I would say it depends on what she is really saying – if it is “my mom is super traditional and expects yuino,” “convince my mom to let me marry you,” or “my mom is fine but just wants to meet you and understand you’re not a common criminal,”
IIRC the Swissotel in Nanba had private rooms for their Japanese restaurant and ANA does as well for some of their restaurants (hard to go wrong with a nice hotel).
Not any specific restaurant, but recommend doing it at a restaurant with a cuisine close to your home country. That way you have a little cultural ”upper hand” 
Maybe try a hotel restaurant at a 5 star or other top hotel?
I did mine over afternoon tea at a fancy hotel… live music and plenty of sweet desserts and most importantly champagne… it really got her mother in a good mood so I highly recommend
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You may also want to try asking in /r/Osaka.
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I wouldn’t do that in public personally speaking.
XEX West in Umeda. Super fancy and super pricey. Took my wife here for our anniversary last year. You can also make a reservation on line
I would say it depends on what she is really saying – if it is “my mom is super traditional and expects yuino,” “convince my mom to let me marry you,” or “my mom is fine but just wants to meet you and understand you’re not a common criminal,”
IIRC the Swissotel in Nanba had private rooms for their Japanese restaurant and ANA does as well for some of their restaurants (hard to go wrong with a nice hotel).
Not any specific restaurant, but recommend doing it at a restaurant with a cuisine close to your home country. That way you have a little cultural ”upper hand” 
Maybe try a hotel restaurant at a 5 star or other top hotel?
(Not the buffet restaurant of course)
Le Saïzeryá
Osaka Asai https://www.ichicore.co.jp/
One of the best restaurants in Osaka
I did mine over afternoon tea at a fancy hotel… live music and plenty of sweet desserts and most importantly champagne… it really got her mother in a good mood so I highly recommend
Modo di Ponte Vecchio
06-4795-8888
https://goo.gl/maps/vsH5qVeSsYxnvWvf7
Has private rooms that can be reserved for 500 yen, if I remember correctly.
Che’hidakaya le ramenerie
Swissotel nankai, top floor fancy restaurant. Online reservations