In order to open the qualifying tournament, you must do some tournament first. To make is simple, tournament that requires cosplay parts or specific shinki is not important. Only the tag, normal and weapon limit tournament that are required for.
For weapon limit restrictions, here is the copy from page 40 to help you out.
Hope it can help.Melee type are
ナックル = Knuckle
ダブルナイフ = Dual Dagger
ドリル = Drill
パイルバンカー = Pile Bunker
小剣 = 1h sword
ロッド = Rod
槍 = Spear
ダブルブレード = Dual Blade
大剣 = 2h Sword
斧 = Axe
ハンマー = Hammer
楽器 = Musical Instrument
Range type are
ハンドガン = Handgun
機関銃 = Machine gun
ライフル = Rifle
ショットガン = Shotgun
ガトリング = Gatling
ランチャー = Launcher
バズーカ = Bazooka
ミサイル = Missile
ビット = Bit
粒子ブラスター = Particle Blaster
浮遊機雷 = Air Mine
There are also property limit
火器 = Firearms
光学 = Optics
As long you enter with the word mention in the battle list, it will be fine.
Example: If the battle description has these conditions
火器 3%
ミサイル/ビット
that means you can only enter with missile or bit weaponry with at least 3% in firearms property.