Searching through strings in a dataframe and increasing the numbers found by 1

I have a dataframe that I have created by hand. I am working on a code that copies the dataframe and concatenates the new dataframe to the end of the first one. For now, I need the code to look through each value of a column of the 'Name' dataframe that contains strings and if there is a number in the string, increase this number by 1. I need the number to be turned into an int so that I can create a function that will look through the dataframe and automatically add 1 to the largest number in the dataframe. An example:

import pandas as pd
data = {'ID': [1,2,3,4],
        'Name': ['BN #1', 'HHC', 'A comp', 'B Comp']}
df = pd.DataFrame(data)

df['SysNum'] = [int(re.search('(?<=#)\d', x)[0]) for x in df['Name'].values]

Afterwards the new df looks like

data2 = {'ID': [1,2,3,4,5,6,7,8],
        'Name': ['BN #1', 'HHC', 'A comp', 'B Comp','BN #2', 'HHC', 'A comp', 'B Comp']} 

When I run this, I receive a 'NoneType' object is not subscriptable error. This makes sense because only the BN # row has a number and re.search returns None when the string parameters are not met, but I cannot figure out how to tell python to ignore the other rows.

EDIT Only the first row each dataframe will increase by 1, so if there is an easier way where I do not use re.search, that is fine. I know there are a couple ways of doing this but I want to be able to always look through the string value of BN and increase it by 1 every time I run the code.

REGEX EDIT

    df2['BaseName'] = [re.sub('\d', '', x) for x in df2['Name'].values]
    df['BaseName'] = [re.sub('\d', '', x) for x in df['Name'].values]
    df2['SysNum'] = [int(re.search('(?<=#)\d', x)[0]) for x in df2['Name'].values]
    # df2['SysNum'] = df2['Name'].get(r'(?<=#)\d').astype(int)
    # df['SysNum'] = [int(re.search('(?<=#)\d', x)[0]) for x in df['Name'].values]
    df['SysNum'] = df['Name'].str.contains('(?<=#)\d').astype(int)
    
    m =  re.search(r'(?<=#)\d', df2['Name'].iloc[0])
    if m:
        df2['SysNum'] = int(m.group(0)) + 1
    
    n = re.search(r'(?<=#)\d', df['Name'].iloc[0])
    if n:
        df['SysNum'] = int(n.group(1)) + 1
    
    
    new_names = df2['BaseName'].unique()
    maxes2 = np.zeros((len(new_names), ))
    
    for j in range(len(new_names)):
        un2 = new_names[j]
        maxes2[j] = df['SysNum'].loc[df['BaseName'] == un2].max()
        df2['SysNum'].loc[df2['BaseName'] == un2] = np.linspace(1, len(df2['SysNum'].loc[df2['BaseName'] == un2]), len(df2['SysNum'].loc[df2['BaseName'] == un2]))
        df2['SysNum'].loc[df2['BaseName'] == un2] += maxes2[j]
        newnames2 = [s + '%d' % num for s,num in zip(df2['BaseName'].loc[df2['BaseName'] == un2].values, df2['SysNum'].loc[df2['BaseName'] == un2].values)]
        df2['Name'].loc[df2['BaseName'] == un2] = newnames2

I have this code working for two dataframes and the numbering works out how I would like it to. The first two have a "Name-###" naming convention for all the rows in the dataframe. This allows the commented out re.search line at the top to run just fine. The next two dataframes I am working on are like the examples I put up earlier with the BN #1 and the rest of the names do not have a number. When I run the commented out re.search lines, the code tries to convert the NoneTypes to int and it cannot do that. When I run the code as is now, a new number is put on each and every row immediately following the name, but I need it to add a new number to the row with the #. So what I need and I am struggling with is a piece of code that looks through the dataframe, looks for a # sign, turns the number after the # sign into an int, a loop that looks for the max int and then adds 1 to that number, adds that new number onto the new dataframe, adds new dataframe onto the old one for a larger master list.