I think this is another good candidate for romanization, as it's a specific class of onsen that a tourist should be able to identify. What fun is it to travel abroad and not learn the local names of the new things you experience?

Also, there's apparently a law that defines "gensen kakenagashi" and other such onsen-related terms, so I think that the label is meaningful and should be retained with precision in English ("kakenagashi" by itself has a different meaning).

You might have to rearrange the sentence a little bit in order to avoid awkward phrasing, but I imagine the second phrase (源泉かけ流しの大きな手湯場) would go something like "a large, "gensen kakenagashi"-style hand bath, with water flowing continuously and directly from the hot spring source."

http://www.hitou.or.jp/en/spateaching/spateaching01.html
In Japan, a law was created known as the 'onsenhou' (Hot Spring Law). Based on this law, a natural spring that contains over a defined amount of natural chemical components, and is over a temperature of 25 degrees at its point of release, is officially considered an 'onsen'.
There are a number of different onsen categories, according to chemical composition and temperature.

We, the HITOU - Japan Association of Secluded Hot Spring Inns continue to study about the best ways to ensure onsen are well managed and moderated, and that our earth's resources are carefully protected and preserved, while endeavouring to make more information available about the various classifications and categories of onsen (sources with a continuous flow, or onsen that are circulated etc.)

1. Continuous flow from source (Gensen kakenagashi)
Whilst heat might be added, there is no water added to this type of bath. With this kind of system, the high temperatures at the source are made more moderate and suitable for bathing by the use of heat-exchange devices either at the point where the water enters the bath, or in the bath itself.
The hot spring of a 'Gensen kakenageshi' bath is always released without being used in a circulation system.

2. Continuous flow (Kakenagashi)
In order to maintain the moderate temperature of this type of bath, both water and heat are added to the water supply. Like the 'Gensen' varient of 'Kakenagashi' - once the water enters the bath, it is released and does not go through a circulation system.

http://www.osaka-info.jp/en/search/list/sightseeing.php?p=19...
This bijin-no-yu, direct from the source (gensen kakenagashi) is sure to heal the tired body and soul.

This is not the best English, but it echoes the idea that the Japanese name is important:
http://www1.town.kusatsu.gunma.jp/etownsys/keitai/330_MEISAI...
This abundant supply of water makes this style of hot spring luxurious that never needs reheating, the so-called "Gensen Kakenagashi" which means the water comes directly from the hot spring source.