So when using the default driver, your wlan interface came up as wlp2s0b1. After the bcmwl driver has been installed, the interface comes up instead as wlp2s0.
So this new interface doesn't match the configuration that NetworkManager was using before, and so you do not automatically reconnect after disconnect (the disconnect itself is unavoidable when changing driver).
I think this is probably not sanely fixable, short of imposing some additional rules to force the network device name to be the same in the two drivers.
The key part of dmesg is:
[ 214.287602] wlp2s0b1: authenticate with 74:da:88:50:00:47 bss_info_ changed: associated bss_info_ changed: qos enabled: true (implement) d11hdrs_ mac80211: txop exceeded phylen 159/256 dur 1778/1504 d11hdrs_ mac80211: txop exceeded phylen 137/256 dur 1602/1504 NETDEV_ CHANGE) : wlp2s0b1: link becomes ready bss_info_ changed: arp filtering: 1 addresses (implement) bss_info_ changed: disassociated bss_info_ changed: arp filtering: 1 addresses (implement) bss_info_ changed: qos enabled: false (implement)
[ 214.292003] wlp2s0b1: send auth to 74:da:88:50:00:47 (try 1/3)
[ 214.294069] wlp2s0b1: authenticated
[ 214.295333] wlp2s0b1: associate with 74:da:88:50:00:47 (try 1/3)
[ 214.299380] wlp2s0b1: RX AssocResp from 74:da:88:50:00:47 (capab=0x431 status=0 aid=2)
[ 214.299928] brcmsmac bcma0:1: brcmsmac: brcms_ops_
[ 214.299932] brcmsmac bcma0:1: brcms_ops_
[ 214.299942] wlp2s0b1: associated
[ 214.331657] brcmsmac bcma0:1: wl0: brcms_c_
[ 214.371947] brcmsmac bcma0:1: wl0: brcms_c_
[ 214.408585] IPv6: ADDRCONF(
[ 214.439835] brcmsmac bcma0:1: brcms_ops_
[ 491.393622] wlp2s0b1: deauthenticating from 74:da:88:50:00:47 by local choice (Reason: 3=DEAUTH_LEAVING)
[ 491.396079] brcmsmac bcma0:1: brcmsmac: brcms_ops_
[ 491.396086] brcmsmac bcma0:1: brcms_ops_
[ 491.396089] brcmsmac bcma0:1: brcms_ops_
[ 491.472322] wl: module verification failed: signature and/or required key missing - tainting kernel
[ 491.516220] wl 0000:02:00.0 wlp2s0: renamed from wlan0
[ 491.596460] wlp2s0: Broadcom BCM4727 802.11 Hybrid Wireless Controller 6.30.223.271 (r587334)
So when using the default driver, your wlan interface came up as wlp2s0b1. After the bcmwl driver has been installed, the interface comes up instead as wlp2s0.
So this new interface doesn't match the configuration that NetworkManager was using before, and so you do not automatically reconnect after disconnect (the disconnect itself is unavoidable when changing driver).
I think this is probably not sanely fixable, short of imposing some additional rules to force the network device name to be the same in the two drivers.