Unfortunately, Chase business cards are out because of the IHG card. For all Chase business cards, you need to have had 0 applications in the last 30 days.
I'm thinking about the
Amex Platinum Business card if you can get a 100k+ Membership Rewards points offer. The big reason is the Amex Platinum Business allows you to earn 1.5x MR on charges of $5,000+. This will also get your toes wet into the world of MR, which can be very useful for booking airline travel, whether by transferring points to partners or redeeming points for paid flights with a 35% rebate on the Amex Platinum Business.
The standard offer is 75k MR points (broken up into 50k MR after $10k, an additional 25k MR after another $10k in 3 months). This public offer is terrible because of the high spend, low rewards. Unfortunately, higher offers on Amex business cards are targeted to accounts, by mailers, or online ads; and although /r/churning subreddit has a code sharing thread that's supposed to share mailer RSVP codes, posters often ask for compensation in return. DoC has posts
here and
here about how to call into Amex to request the higher offers (for Amex Platinum Business, it's usually 100k MR after $10k/3 months, which is perfect for your situation). Of course, if you've been conservative about applications, you'll have to toughen yourself up and get some numbers and a story straight to apply for a business card over the phone.
The Amex Platinum Business does carry a hefty $450 AF, but it comes with $200 in airline fee credits that can be redeemed for gift cards with Southwest, AA, and Delta (in certain denominations per transaction); $100 Global Entry or $85 TSA Pre✓ credit; automatic Gold status at Hilton, SPG/Marriott; access to Amex Centurion Lounges at DFW, IAH, LAS, LGA, PHL, MIA, SEA, SFO, and Priority Pass lounges (although CSR's access is better).
If you're not comfortable with the AF on the Amex Platinum Business, you could use the above method to apply for the
Amex Business Rewards Gold with a 75k MR offer after $10k/3 months (check the above DoC link for details). This card carries a $175 AF, which is waived in the first year.
Neither card will report to your personal credit report, so you won't be burning a */24 slot on either.
ETA: a few additional thoughts -- before applying for an Amex card, make sure your bill accepts Amex cards directly or the bill is of a type that you can use Plastiq to send a payment. Also, if you have until the end of this month to pay the bill, there might be a small window to get a CIP or CIC at the end of April as long as it's been more than 30 days since your IHG application, then ask Chase to expedite the card to you. This may be risky because business applications can go pending for a few days before approval.