der Bausch
wad
A soft, shapeless, bunched-up amount of a material such as cotton wool, fabric, paper, or wool.
Die Ärztin reinigte die Wunde mit einem Bausch aus Verbandsmull.
The doctor cleaned the wound with a wad of gauze.

wad / lump
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wad
A soft, shapeless, bunched-up amount of a material such as cotton wool, fabric, paper, or wool.
Die Ärztin reinigte die Wunde mit einem Bausch aus Verbandsmull.
The doctor cleaned the wound with a wad of gauze.

The expression 'in Bausch und Bogen' is a fixed idiom. The words 'Bausch' and 'Bogen' are never used here alone or in a different order.
Sein Vorschlag wurde in Bausch und Bogen verworfen.
His proposal was rejected outright.
The idiom 'in Bausch und Bogen' is almost always combined with negative verbs such as 'ablehnen' (to reject), 'verwerfen' (to dismiss/reject), or 'verdammen' (to condemn) and means 'vollständig' (completely) or 'komplett' (entirely).
Die Opposition lehnte das neue Gesetz in Bausch und Bogen ab.
The opposition rejected the new law outright.
In its literal meaning, 'Bausch' (wad) is often combined with soft, moldable materials. Typical phrases are 'ein Bausch Watte' (a wad of cotton wool), 'ein Bausch Stoff' (a wad of fabric), or 'ein Bausch Papier' (a wad of paper).
Sie tupfte die Farbe mit einem Bausch aus alten Tüchern auf die Wand.
She dabbed the paint onto the wall with a wad of old cloths.
complete
'vollständig' and 'in Bausch und Bogen' can both express that something happens completely and without exception. 'vollständig' is a normal adjective or adverb and fits in many situations: a form can be complete, or something can disappear completely. 'in Bausch und Bogen', on the other hand, is a fixed expression. It sounds stronger and is used almost only with verbs like 'ablehnen' (to reject), 'verwerfen' (to dismiss), or 'verdammen' (to condemn). So you use it especially when someone rejects something completely. For neutral situations, 'vollständig' is more natural.

Der Vorschlag wurde in der Sitzung in Bausch und Bogen abgelehnt.
The proposal was rejected outright at the meeting.
Nach dem Sturm war der kleine Weg im Park vollständig mit Ästen bedeckt.
After the storm, the small path in the park was completely covered with branches.